Math 208
Let A=−2 3 4−6, B=10 4 8 1, and C=4 −2 4 −3. Verify that AB=AC and yet B≠C. Question content area bottom Part 1 Show the calculations that are used to find the entries for matrix AB. Choose the correct answer below. Part 2 Show the calculations that are used to find the entries for matrix AC. Choose the correct answer below.
1.) −2(10)+3(8)−2(4)+3(1) 4(10)+(-6)(8)4(4)+(-6)(1) 2. −2(4)+3(4)−2(−2)+3(−3)4(4)+(−6)(4)4(−2)+(−6)(−3)
Use the given inverse of the coefficient matrix to solve the following system. 5x1+3x2=12 −6x1−3x2=3 A−1= −1 −1 2 5/3
=−15 and x2=29
An m×n lower triangular matrix is one whose entries above the main diagonal are zeros, as is shown in the matrix to the right. When is a square lower triangular matrix invertible? Justify your answer. 3000 4100 7420 4681
A square lower triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the n×n matrix has n pivot positions.
An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer. 3474 0146 0028 0001
A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the n×n matrix has n pivot positions.
Let W be the union of the second and fourth quadrants in the xy-plane. That is, let W=xy : xy≤0. If u is in W and c is any scalar, is cu in W? Why?
If u=xy is in W, then the vector cu=cxy=cxcy is in W because (cx)(cy)=c2(xy)≤0 since xy≤0.
Explain why the columns of an n×n matrix A are linearly independent when A is invertible.
If A is invertible, then the equation Ax=0 has the unique solution x=0. Since Ax=0 has only the trivial solution, the columns of A must be linearly independent.
State which property of determinants is illustrated in this equation. −8 2 6 −24 −3 −8 −9 −6 9 =−8 2 6 0 −9 −26 −9 −6 9 .
If a multiple of one row of A is added to another row to produce matrix B, then det B=det A.
State which property of determinants is illustrated in this equation. −4−9−5 12−5−3 −7 4−8 =−1 2−5−3 −4−9−5 −7 4−8
If two rows of A are interchanged to produce B, then det B=−det A. Your
If A is invertible, then the columns of A−1 are linearly independent. Explain why.
It is a known theorem that if A is invertible then A−1 must also be invertible. According to the Invertible Matrix Theorem, if a matrix is invertible its columns form a linearly independent set. Therefore, the columns of A−1 are linearly independent.
If C is 6×6 and the equation Cx=v is consistent for every v in ℝ6, is it possible that for some v, the equation Cx=v has more than one solution? Why or why not?
It is not possible. Since Cx=v is consistent for every v in ℝ6, according to the Invertible Matrix Theorem that makes the 6×6 matrix invertible. Since it is invertible, Cx=v has a unique solution.
Is it possible for a 5×5 matrix to be invertible when its columns do not span ℝ5? Why or why not?
It is not possible; according to the Invertible Matrix Theorem an n×n matrix cannot be invertible when its columns do not span ℝn.
Use matrix algebra to show that if A is invertible and D satisfies AD=I, then D=A−1.
Left-multiply each side of the equation AD=I by A−1 to obtain A−1AD=A−1I, ID=A−1, and D=A−1.
The equation below illustrates a property of determinants. State the property. [3 −6 9 3 5 −5 1 3 3] =3 [1 −2 3 3 5 −5 1 3 3]
Multiplying a row by 3 multiplies the determinant by 3.
Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant. −3 6 −4 1 1 1 5− 2 2, −3 6 −4 k k k 5 −2 2 What is the elementary row operation? How does the row operation affect the determinant?
Replace row 2 with k times row 2. The determinant is multiplied by k.
Consider the accompanying matrix as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system. 1 −6 4 0 −2 0 2 −6 0 5 0 0 1 4 −3 0 0 4 13 −3 What should be the first elementary row operation performed? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Replace row 4 by its sum with −4 times row 3.
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant. 217 594 abc, 217 abc 594 What is the elementary row operation? How does the row operation affect the determinant?
Rows 2 and 3 are interchanged. It changes the sign of the determinant.
What should be the second elementary row operation performed? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Scale row 4 by −1/3.
Explain why the columns of an n×n matrix A span ℝn when A is invertible.
Since A is invertible, for each b in ℝn the equation Ax=b has a unique solution. Since the equation Ax=b has a solution for all b in ℝn, the columns of A span ℝn.
Suppose AB=AC, where B and C are n×p matrices and A is invertible. Show that B=C. Is this true, in general, when A is not invertible? What can be deduced from the assumptions that will help to show B=C?
Since matrix A is invertible, A−1 exists.
Suppose A, B, and X are n×n matrices with A, X, and A−AX invertible, and suppose that (A−AX)−1=X−1B. Explain why B is invertible. Choose the correct answer below.
Solving the equation (A−AX)−1=X−1B for B yields X(A−AX)−1=B. Since X is invertible and (A−AX)−1 is invertible, the product X(A−AX)−1=B is also invertible.
Suppose A is n×n and the equation Ax=0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In.
Suppose A is n×n and the equation Ax=0 has only the trivial solution. Then there are no free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the n×n identity matrix, In.
If the given equation Gx=y has more than one solution for some y in ℝn, can the columns of G span ℝn? Why or why not? Assume G is n×n.
The columns of G cannot span ℝn. According to the Invertible Matrix Theorem, if Gx=y has more than one solution for some y in ℝn, that makes the matrix G non invertible.
If an n×n matrix K cannot be row reduced to In, what can you say about the columns of K? Why?
The columns of K are linearly dependent and the columns do not span ℝn. According to the Invertible Matrix Theorem, if a matrix cannot be row reduced to In that matrix is non invertible.
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let A=3 0 −3 3 −5 3, B=6 −5 2 2−4−4 C=2 3 −2 2 and D=4 5 −1 3. −2A, B−2A, AC, CD Compute the matrix product AC. Select the correct choice below and, if necessary, fill in the answer box within your choice.
The expression AC is undefined because the number of columns in A is not equal to the number of rows in C.
Suppose the first two columns, b1 and b2, of B are equal. What can you say about the columns of AB (if AB is defined)? Why?
The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal.
Find the inverse of the matrix. 2 9 6 8
The inverse matrix is −4/19 9/38 3/19 −1/19.
Find the inverse of the given matrix, if it exists. A=1 −2 1 4 −7 3 −2 6 −4
The matrix A does not have an inverse.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. 5 6 -4 -2
The matrix is invertible because its determinant is not zero.
Use determinants to find out if the matrix is invertible. 10 5 1 2 −15 2 0 25 −3
The matrix is invertible because the determinant of the matrix is not zero.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. 3474 0136 0029 0001
The matrix is invertible. The given matrix has 4 pivot positions.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. 3 0 0 −5 −6 0 5. 7 −1
The matrix is invertible. The given matrix has three pivot positions.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. −8 −2 4 1
The matrix is not invertible because its determinant is zero.
Use determinants to find out if the matrix is invertible. 5 0 −1 2 −6 −4 0 10 6
The matrix is not invertible.
Can a square matrix with two identical columns be invertible? Why or why not?
The matrix is not invertible. If a matrix has two identical columns then its columns are linearly dependent. According to the Invertible Matrix Theorem this makes the matrix not invertible.
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. 3 0 -4 2 0 5 -3 0 8
The matrix is not invertible. If the given matrix is A, the columns of A do not form a linearly independent set.
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 1 2 -5 4 h -20
The matrix is the augmented matrix of a consistent linear system for every value of h.
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 8 14 h -4 -7 3
The matrix is the augmented matrix of a consistent linear system if h=−6.
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 1 h 5 3 6 10
The matrix is the augmented matrix of a consistent linear system if h≠2.
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. 1 h 4 -5 10 -24
The matrix is the augmented matrix of a consistent linear system if h≠−2.
State the row operation performed below and describe how it affects the determinant. ab cd, cd ab Question content area bottom Part 1 What row operation was performed? How does this affect the determinant?
The row operation swaps rows 1 and 2. The sign of the determinant is reversed.
Determine if the given set is a subspace of ℙ5. Justify your answer. The set of all polynomials of the form p(t)=at5, where a is in ℝ.
The set is a subspace of ℙ5. The set contains the zero vector of ℙ5, the set is closed under vector addition, and the set is closed under multiplication by scalars.
Use determinants to decide if the set of vectors is linearly independent. 6 4 -4 0 10 -12 -2 0 -6 0 3 0 5 8 0 -2
The set of vectors is linearly dependent.
Use determinants to decide if the set of vectors is linearly independent. -7 6 -7 5 -4 0 -3 -7 4 Is the set of vectors linearly independent?
The set of vectors is linearly independent, because the determinant is not zero.
If A, B, and C are n×n invertible matrices, does the equation C−1(A+X)B−1=In have a solution X? If so, find it. Question content area bottom Select the correct choice below and, if necessary, fill in the answer box within your choice.
The solution is X=CB minus Upper ACB−A.
The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system. 1−290 0170 0060 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The solution set contains one solution, (0,0,0).
The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system. 1−300−4 01−10−6 001−22 00014
The solution set contains one solution: (8,4,10,4).
The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system. 000−3 01−14 0013 172−3 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The solution set is empty.
Determine whether the statement below is true or false. Justify the answer. A 5×6 matrix has six rows.
The statement is false. A 5×6 matrix has five rows and six columns.
Determine whether the statement below is true or false. Justify the answer. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
The statement is false. If A=2613, then det A=0 and the rows and columns are all distinct and not full of zeros.
Determine whether the statement below is true or false. Justify the answer. If A=abcd and ab−cd≠0, then A is invertible.
The statement is false. If ad−bc≠0, then A is invertible.
Determine whether the statement below is true or false. Justify the answer. A product of invertible n×n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
The statement is false. If A and B are invertible matrices, then (AB)−1=B−1A−1.
Determine whether the statement below is true or false. Justify the answer. If three row interchanges are made in succession, then the new determinant equals the old determinant.
The statement is false. If three row interchanges are made in succession, then the new determinant equals the negative of the old determinant.
Let A, B, and C be arbitrary matrices for which the indicated products are defined. Determine whether the statement below is true or false. Justify the answer. (AB)C=(AC)B
The statement is false. The associative law of multiplication for matrices states that A(BC)=(AB)C.
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
The statement is false. The definition of AB states that each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. The determinant of a triangular matrix is the sum of the entries on the main diagonal.
The statement is false. The determinant of a triangular matrix is the product of the entries along the main diagonal.
Determine whether the statement below is true or false. Justify the answer. The echelon form of a matrix is unique.
The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.
Determine whether the statement below is true or false. Justify the answer. If A and B are n×n and invertible, then A−1B−1 is the inverse of AB.
The statement is false. The inverse of AB is B−1A−1.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the linear transformation x ↦ Ax maps ℝn into ℝn, then A has n pivot positions.
The statement is false. The linear transformation x ↦ Ax will always map ℝn into ℝn for any n×n matrix. According to the Invertible Matrix Theorem, A has n pivot positions only if x ↦ Ax maps ℝn onto ℝn.
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. If A and B are 3×3 matrices and B=b1b2b3, then AB=Ab1+Ab2+Ab3.
The statement is false. The matrix Ab1+Ab2+Ab3 is a 3×1 matrix, and AB must be a 3×3 matrix. The plus signs should be spaces between the 3 columns.
Determine whether the statement below is true or false. Justify the answer. The transpose of a product of matrices equals the product of their transposes in the same order.
The statement is false. The transpose of a product of matrices equals the product of their transposes in the reverse order.
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. (AB)T=ATBT
The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or (AB)T=BTAT.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. The determinant of A is the product of the diagonal entries in A.
The statement is false. This is only true if A is triangular.
Determine whether the statement below is true or false. Justify the answer. Two matrices are row equivalent if they have the same number of rows.
The statement is false. Two matrices are row equivalent if there exists a sequence of elementary row operations that transforms one matrix into the other.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If there is a b in ℝn such that the equation Ax=b is inconsistent, then the transformation x ↦ Ax is not one-to-one.
The statement is true. According to the Invertible Matrix Theorem, if there is a b in ℝn such that the equation Ax=b is inconsistent, then equation Ax=b does not have at least one solution for each b in ℝn and this makes A not invertible.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If AT is not invertible, then A is not invertible.
The statement is true. By the Invertible Matrix Theorem, if AT is not invertible, then all statements in the theorem are false, including A is invertible. Therefore, A is not invertible.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the equation Ax=b has at least one solution for each b in ℝn, then the solution is unique for each b.
The statement is true. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in ℝn, then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the columns of A are linearly independent, then the columns of A span ℝn.
The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A must span ℝn.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
The statement is true. By the Invertible Matrix Theorem, if the equation Ax=0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n×n identity matrix.
The statement is true. By the Invertible Matrix Theorem, if the equation Ax=0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the n×n identity matrix.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If there is an n×n matrix D such that AD=I, then there is also an n×n matrix C such that CA=I.
The statement is true. By the Invertible Matrix Theorem, if there is an n×n matrix D such that AD=I, then it must be true that there is also an n×n matrix C such that CA=I.
Determine whether the statement below is true or false. Justify the answer. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
The statement is true. Each elementary row operation replaces a system with an equivalent system.
Determine whether the statement below is true or false. Justify the answer. If A=abcd and ad=bc, then A is not invertible.
The statement is true. If ad=bc then ad−bc=0, and 1/ad−bc d −b −c a is undefined.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. If the columns of A are linearly dependent, then det A=0.
The statement is true. If the columns of A are linearly dependent, then A is not invertible.
Determine whether the statement below is true or false. Justify the answer. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
The statement is true. It is the definition of a basic variable.
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. The second row of AB is the second row of A multiplied on the right by B.
The statement is true. Let rowi(A) denote the ith row of matrix A. Then rowi(AB)=rowi(A)B. Letting i=2 verifies this statement.
Determine whether the statement below is true or false. Justify the answer. Reducing a matrix to echelon form is called the forward phase of the row reduction process.
The statement is true. Reducing a matrix to echelon form is called the forward phase and reducing a matrix to reduced echelon form is called the backward phase.
Determine whether the statement below is true or false. Justify the answer. If A is invertible, then the inverse of A−1 is A itself.
The statement is true. Since A−1 is the inverse of A, A−1A=I=AA−1. Since A−1A=I=AA−1, A is the inverse of A−1.
Determine whether the statement below is true or false. Justify the answer. If A can be row reduced to the identity matrix, then A must be invertible.
The statement is true. Since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible.
Determine whether the statement below is true or false. Justify the answer. If A is an invertible n×n matrix, then the equation Ax=b is consistent for each b in ℝn.
The statement is true. Since A is invertible, A−1b exists for all b in ℝn. Define x=A−1b. Then Ax=b.
For this exercise assume that the matrices are all n×n. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the columns of A span ℝn, then the columns are linearly independent.
The statement is true. The Invertible Matrix Theorem states that if the columns of A span ℝn, then matrix A is invertible. Therefore, the columns are linearly independent.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices.
The statement is true. The determinant of an n×n matrix A can be computed by a cofactor expansion across any row or down any column. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size (n−1)×(n−1).
Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Determine whether the statement below is true or false. Justify the answer. AB+AC=A(B+C)
The statement is true. The distributive law for matrices states that A(B+C)=AB+AC.
Determine whether the statement below is true or false. Justify the answer. In order for a matrix B to be the inverse of A, both equations AB=I and BA=I must be true.
The statement is true. The product of a matrix and its inverse is the identity matrix.
Determine whether the statement below is true or false. Justify the answer. A general solution of a system is an explicit description of all solutions of the system.
The statement is true. The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system.
Let A and B be arbitrary matrices for which the indicated sum is defined. Determine whether the statement below is true or false. Justify the answer. A^T+B^T=(A+B)^T
The statement is true. The transpose property states that (A+B)T=AT+BT.
Determine whether the statement below is true or false. Justify the answer. Two fundamental questions about a linear system involve existence and uniqueness.
The statement is true. The two fundamental questions are about whether the solution exists and whether there is only one solution.
Determine whether the statement below is true or false. Justify the answer. Every elementary row operation is reversible.
The statement is true. Replacement, interchanging, and scaling are all reversible.
Suppose H is an n×n matrix. If the equation Hx=c is inconsistent for some c in ℝn, what can you say about the equation Hx=0? Why?
The statement that Hx=c is inconsistent for some c is equivalent to the statement that Hx=c has no solution for some c. From this, all of the statements in the Invertible Matrix Theorem are false, including the statement that Hx=0 has only the trivial solution. Thus, Hx=0 has a nontrivial solution.
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
Do the three planes x1+4x2+x3=4,x2−x3=1, and 3x1+15x2=8 have at least one common point of intersection? Explain.
The three planes do not have a common point of intersection.
Solve the system. x1 −6x3 =22 4x1+2x2−9x3=49 x2+5x3=−12
The unique solution of the system is (4, 3, −3).
Let V be the set of vectors shown below. V=xy : x>0, y≤0 a. If u and v are in V, is u+v in V? Why? b. Find a specific vector u in V and a specific scalar c such that cu is not in V.
The vector u+v must be in V because the x-coordinate of u+v is the sum of two positive numbers, which must also be positive, and the y-coordinate of u+v is the sum of nonpositive numbers, which must also be nonpositive. u=2−2, c=−1
Suppose a 3×8 coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have nine columns and will not have a row of the form [000000001], so the system is consistent.
Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. 3 0 3 2 3 2 0 5 −1 Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice. Compute the determinant using a cofactor expansion down the second column. Select the correct choice below and fill in the answer box to complete your choice.
Using this expansion, the determinant is (3)(−13)−(0)(−2)+(3)(10) =negative 9−9. Using this expansion, the determinant is −(0)(−2)+(3)(−3)−(5)(0) =negative 9−9.
Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column. 4 −4 5 5 1 4 1 5 −1 Write the expression for the determinant using a cofactor expansion across the first row. Choose the correct answer below. Write the expression for the determinant using a cofactor expansion down the second column. Choose the correct answer below.
Using this expansion, the determinant is (4)(−21)−(−4)(−9)+(5)(24). Using this expansion, the determinant is −(−4)(−9)+(1)(−9)−(5)(−9). Your answer is correct. Part 3 The determinant is 0.
Determine which matrices are in reduced echelon form and which others are only in echelon form. a. 1000 0100 0011 b. 01111 00111 00001 00000 c. 0000 1400 0010 0001
a. reduced b. echelon only c. neither
Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system. 1 -5 6 g 0 9 -9 h -3 6 -9 k
g+h+k=0
Let A=3 2 −2 1 and B= 1 3 −3 k What value(s) of k, if any, will make AB=BA?
k=−2
Find the general solution of the system whose augmented matrix is given below. 3− 7 4 0 9 −21 12 0 6 −14 8 0
x1=7/3x2−4/3x3 x2 is free x3 is free