Matrix Operations LO

Invertible Linear Transformation

A linear transformation T: Rn to Rn is said to be invertible if there exists a function S: Rn to Rn such that S(T (x)) = x for all x in Rn (1) T (S(x)) = x for all x in Rn (2) The next theorem shows that if such an S exists, it is unique and must be a linear transformation. We call S the inverse of T and write it as T ^-1

Nonsingular Matrix

A matrix that IS invertible

Singular Matrix

A matrix that isnt invertible

Theorem 7: Elementary Row Operations and Matrix inversion

An n n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A1.

Another View of Matrix Multiplication

Denote the columns of In by e1; : : : ; en. Then row reduction of [ A I ] to [ I A1 ] can be viewed as the simultaneous solution of the n systems Ax = e1; Ax = e2; : : : ; Ax = en (2) where the "augmented columns" of these systems have all been placed next to A to form [ A e1 e2 en ] = [ A I ] The equation AA1 = I and the definition of matrix multiplication show that the columns of A1 are precisely the solutions of the systems in (2). This observation is useful because some applied problems may require finding only one or two columns of A1. In this case, only the corresponding systems in (2) need to be solved

(T/F) If A and B are 2 x2 with columns a1; a2, and b1; b2, respectively, then AB=[a1b1 a2b2]

FALSE - Not how it works

(T/F) If A and B are 3 x3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3].

FALSE - Not how matrix multiplication works

AB=AC, C must equal B

FALSEE

T/F) If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in Rn

FALSE• A needs to be invertible.

(T/F) (AB)C=(AC)B

False - Matrix Multiplication isnt communative

(T/F) A product of invertible n xn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.

False - Reverse Order

(T/F) If the linear transformation x to Ax maps Rn into Rn, then A has n pivot positions

False A could be any square n × n matrix and not all of them have n pivot positions.

(T/F) Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.

False, they are a linear combination of the columns of A using columns of B as weights

(T/F) (AB)^T=A^TB^T

False. It equals B^T A^T

T/F) If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the nxn identity matrix.

False. This is the same as being one-to-one, but to be row equivalent to In, A must also be onto

(T/F) If A and B are nxn and invertible, then A1B1 is the inverse of AB

False: B^-1A^-1

(T/F) If A is invertible, then the elementary row operations that reduce A to the identity In also reduce A^-1 to In.

False; if A is​ invertible, then the row operations required to reduce A to the identity correspond to some product of elementary matrices E sub1 Esub2 Esub3 ••• Esubp.Then the row operations required to reduce A^−1 to the identity would correspond to the product Esubp^−1 ••• Esub3^−1 Esub2^−1 Esub1^−1. Opposite order

Theorem 5: Invertibility and unique solutions

If A is an invertible mxn matrix, then for each b in Rn, the equation Ax=b has the unique solution x=A^-1b

Definition of a Matrix

If A is an mxn matrix, and if B is an nxp matrix with columns b1 through bp, then the product AB is an mxp matrix whose columns are Ab1 through Abp. Each column AB is a linear combination of the columns of A using b as weights

Explain why the columns of an n x n matrix A are linearly independent when A is invertible.

If A is​ invertible, then the equation Ax equals 0 has the unique solution x equal 0 Since Ax equals 0 has only the trivial​ solution, the columns of A must be linearly independent.

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.

Theorem 4 - Invertible Matrices

Let A = {{a.b},{c,d}}. If ad-bc =/ 0, then A is invertible and A^(-1)=(1/(ad-bc)){{d,-b,},{-c,a}} If ad-bc=0, A is not invertible

Invertibility of products

Let A and B be square matrices. If AB = I , then A and B are both invertible, with B = A^-1 and A = B^-1.

Theorem 9: Invertible Transformations and the Transformation Matrix

Let T: Rn to Rn be a linear transformation and let A be the standard matrix for T . Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S(x) = A^-1x is the unique function satisfying both equations

Let A and B be n times n matrices. Show that if AB is invertible so is B.

Let W be the inverse of AB. Then WAB equals I and (WA)B equals I. Therefore, matrix B is invertible by part​ (j) of the IMT.

If AB = the zero matrix, does A or B equal the zero matrix?

No. Not Necessarily

Explain why the columns of an n times n matrix A span set of real numbers Rn when A is invertible.

Since A is​ invertible, for each b in set of real numbers Rn the equation Ax equals b has a unique solution. Since the equation Ax equals b has a solution for all b in set of real numbers Rn, the columns of A span set of real numbers Rn .

(T/F) If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in Rn.

TRUE

T/F) If A is invertible, then the inverse of A^-1 is A itself.

TRUE

T/F) If the columns of A are linearly independent, then the columns of A span Rn.

TRUE

T/F. A^T + B^T = (A+B)^T

TRUE

(T/F) Each elementary matrix is invertible.

TRUE -

AB=BA

TRUE - Do this on the test PLEASE

(T/F) In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.

TRUE By Definition

The transpose of a product of matrices equals the product of their transposes in the reverse order.

TRUE- Fact checked by true american patriots

T/F) If the columns of A span Rn, then the columns are linearly independent

TRUE. If the columns of A span R^n, then the columns are linearly independent. They span R^n due to no free variables and square matrix. If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.TRUE. If the columns of A span R^n, then the columns are linearly independent. They span R^n due to no free variables and square matrix. If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.

(T/F) If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions

TRUE• More vectors (columns) than rows (entries), meaning free variables and therefore infinitely many solutions.

Invertible Matrix Theorem!

The Invertible Matrix Theorem Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false. a. A is an invertible matrix. b. A is row equivalent to the nxn identity matrix. c. A has n pivot positions. d. The equation Ax = 0 has only the trivial solution. e. The columns of A form a linearly independent set. f. The linear transformation x to Ax is one-to-one. g. The equation Ax = b has at least one solution for each b in Rn. h. The columns of A span Rn. i. The linear transformation x to Ax maps Rn onto Rn. j. There is an nxn matrix C such that CA = I . k. There is an nxn matrix D such that AD = I . l. AT is an invertible matrix

Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you say about the columns of​ A?

The columns of A are linearly dependent because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. Since bp is not all​ zeros, then any solution to Abp = 0 can not be the trivial solution.

If the given equation Gx= y has more than one solution for some y in set of real numbers Rn, can the columns of G span set of real numbers Rn? Why or why​ not? Assume G is n times n.

The columns of G cannot span set of real numbers Rn. According to the Invertible Matrix​ Theorem, if Gx=y has more than one solution for some y in set of real numbers Rn, that makes the matrix G non invertible

If an n times 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 set of real numbers Rn .According to the Invertible Matrix​ Theorem, if a matrix cannot be row reduced to In that matrix is non invertible.

Can a square matrix with two identical columns be​ invertible? Why or why​ not?

TheThe 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.

(T/F) The second row of AB is the second row of A multiplied on the right by B

True

T/F. If A= {{a,b},{c,d}} and ab-cd =/0, then A is invertible

True

T/F. If A= {{a,b},{c,d}} and ab=cd, then A is not invertible

True

(T/F): AB + AC = A(B+C)

True - Associative Property

(T/F) If AT is not invertible, then A is not invertible.

True by invertible matrix theorem

T/F) If there is a b in Rn such that the equation Ax = b is inconsistent, then the transformation x to Ax is not one-to-one

True; according to the Invertible Matrix Theorem if there is a b in set of real numbers ℝn such that the equation Ax=b is​ inconsistent, then equation Ax=b does not have at least one solution for each b in set of real numbers ℝn and this makes A not invertible.

T/F) If the equation Ax = b has at least one solution for each b in Rn, then the solution is unique for each b.

True; by the Invertible Matrix Theorem if Ax=b has at least one solution for each b in set of real numbers ℝn​, then matrix A is invertible. If A is​ invertible, then according to the invertible matrix theorem the solution is unique for each b.

T/F) If A can be row reduced to the identity matrix, then A must be invertible.

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.

(T/F) If there is an nxn matrix D such that AD = I , then there is also an nxn matrix C such that CA = I

Truth Brother Inshallah

Properties of matrix multiplication

a) A(BC)=(AB)C - Associative Law b) A(B+C)=AB+AC - Left Distributive Law c) (B+C)A = BA+CA - Right Distributive Law d) r(AB)=(rA)B=A(rB) for any scalar r e) ImA=A=AIn

Theorem 1: Matrix Operations

a) A+B = B+A

Theorem 6: Wacky facts about invertible matrices

a) A^-1^-1=A b) (AB)^-1=B^-1A^-1 c) A^T^-1=A^-1^T

Properties of Matrix Transpositions

a) A^T^T=A b) (A+B)^T = A^T+B^T c) For any scalar r, (rA)^T = rA^T d) (AB)^T=B^TA^T

When is a matrix invertible?

it is invertible if it has an mxn matrix C such that CA=I and AC=I A^-1A=I=AA^-1

Row-column rule for matrices

rowi(AB)=rowi(A)*B