Linear Chapter 2 and 3 Test
Cramer's Rule
A formula for each entry in the solution x of the equation Ax=b when A is an invertible matrix. A method that uses determinants to solve a system of linear equations
Invertible
A matrix A is called invertible if there exists a matrix C such that CA = I
Singular
A matrix that has no inverse
Zero Matrix
A matrix with all zero entries The additive Identity
Cofactor
A number Cij=(−1)^i+j det Aij, called the (i,j)-cofactor of A, where Aij is the sub matrix formed by deleting the ith row and the jth column of A
Nonsingular
An invertible matrix
Assume that F is an nxn matrix. If the equation Fx=y is inconsistent for some y in R^n, what can you say about the equation Fx=0?
By the invertible matrix theorem, if Fx=y is inconsistent for some y, then the equation Fx=0 has nontrivial solutions.
If an nxn matrix G cannot be row reduced to In, what can you say about the columns of G?
By the invertible matrix theorem, if G is not row equivalent to the nxn identity matrix then the columns of G do not span R^n, and they do not form a linearly independent set.
If the columns of a 7x7 matrix D are linearly independent, what can you say about the solutions of Dx=b?
By the invertible matrix theorem, we can say that there is exactly one solution for each b in R^7
True or False? Any linear system of n linear equations in n variables can be solved using Cramers rule
False
True or False? If A and B are n x n then (A+B)(A-B) = A^2-B^2
False
True or False? If A and B are nxn matrices with det(A)=2 and det(B)=3 then det(A+B)=5
False
True or False? If A and B are square and invertible, then AB is invertible and (AB)^-1 = (A^-1)(B^-1)
False
True or False? If A is a 3x3 matrix then det(5A) = 5det(A)
False
True or False? If A is an nxn matrix and det(A)=2 then det(A^3) = 6
False
True or False? If A is invertible, and r does not = 0, then (rA)^-1 = (r)(A^-1)
False
True or False? If A is invertible, then det(A^-1) = det(A)
False
True or False? If AB = C and C has 2 columns, then A has 2 columns.
False
True or False? If AB = I, then A is invertible
False
True or False? If AC = 0, then either A = 0 or C = 0
False
True or False? If B is produced by interchanging two rows of A then det(B)=det(A)
False
True or False? If BC = BD, then C = D
False
True or False? If u and v are in R^2 and det([u v]) = 10, then the area of the triangle in the plane with vertices at 0, u, v is 10
False
True or False? det(-A) = -det(A)
False
True or False? det(A^t) = -det(A)
False
Transpose (of A)
Given an n x m matrix A, A^t is when columns are the corresponding rows of the m x n matrix A.
Symmetric Matrix
If A = A^T then A is symmetric
Left Multiplication (By A)
Multiplication by a matrix on the left by (A)
Right multiplication (by A)
Multiplication by a matrix on the right by (A)
Is it possible for a 4x4 matrix to be invertible when its columns do not span R^4. Why or why not?
No because the invertible matrix theorem says that for a matrix to be invertible, its columns must span R^n.
If A is a 5x5 matrix and the equation Ax=b is consistent for every b in R^5, is it possible for some b to have more than one solution?
No, by the invertible matrix theorem every b must have only one solution. If some b had more than one solution the system would be dependent which cannot happen.
If the equation Cu=v has more than one solution for some v in R^n, can the columns of the nxn matrix span R^n?
No, by the invertible matrix theorem, each b must have only one solution for the columns of the nxn matrix to span R^n
Can a square matrix with 2 identical columns be invertible?
No. By the invertible matrix theorem the columns must form a linearly independent set.
Column-row expansion
The expression of a product AB as a sum of other products:
Determinant
The number (detA) defined inductively by a cofactor expansion along the first row of A. Also, (−1)^r times the product of the diagonal entries in any echelon form U obtained from A by row replacements and r row interchanges (but no scaling operations).
True or False? If 2 rows of a 3x3 matrix A are the same then det(A) = 0
True
True or False? If A and B are m x n then both AB^t and A^tB are defined
True
True or False? If A is a 2x2 matrix with a 0 determinant, then one column of A is the multiple of the other
True
True or False? If A is a 3x3 matrix and if the equation Ax = [ 1; 0; 0] has a unique solution then A is invertible
True
True or False? If A is invertible, then det(A^-1)*det(A) = 1
True
True or False? If AB = BA and if A is invertible then (A^-1)(B) = (B)(A^-1)
True
True or False? If A^3 = 0, then det(A)=0
True
True or False? If B is produced by adding one row of A to a linear combination of the other rows then det(B) = det(A)
True
True or False? If B is produced by multiplying row 3 of A by 5 then det(B)=5det(A)
True
True or False? Left-multiplying a matrix B by a diagonal matrix A, with nonzero entries on the diagonal, scales the rows of B.
True
True or False If an nxn matrix A is invertible then the columns of A^t are linearly independent.
True. By the invertible matrix theorem, if a matrix A is invertible then A^t is also invertible, therefore its columns must be independent.
Square Matrix
When the number of rows = the number of columns
If nxn matrices E and F have the property EF=I, do E and F commute?
Yes because one is the inverse of the other and it does not matter if you left multiply or right multiply a matrix by its inverse, you will always obtain I.
True or False? det(A^t*A) is greater than or equal to 0
true
