For all m, n, k e N, all m × n matrices A and all n x k matrices B, C prove that A (B+C) – A ·B+ A · C -

Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is as follows

Q: Prove the formula for a nonsingular n × n matrix A. Assume n ≥ 2.adj[adj(A)] = ∣A∣n−2A

A: We  start with the fundamental equation: A.adj(A) = |A|In

Q: Prove that if B is a 3×1 matrix and C is a 1×3 matrix, then the 3×3 matrix BC has rank at most 1.…

A: Let A be the 3 × 3 matrix of rank 1, then it will be in the form as follows.

Q: Find a 3 x 3 matrix A such that Ax = 9x for all x in R3. A II

A:

Q: Prove that if A is an n × n invertible matrix, then adj(A−1) = [adj(A)]−1.

A: By definition of A-1, we have

Q: Prove using elements that for any square matrices A,B of the same size and constant K.…

A: For any square matrices A,B of the same size and constant K.  tr(kA+B)=ktr(A)+tr(B)

Q: .

A:

Q: Let A be an nxn matrix, then for any r€ R, det(rA) = det A. Select one: O True O False

A: We have to state TRUE or FALSE for the given question.

Q: Let A and B be n × n matrices. Prove that if Ax = B x for all n × 1 matrices x, then A = B .

A: Here given A and B be n × n matrices if x is the matrix of order n × 1,  Then Ax, Ay is also…

Q: Let = Find a nonsingular matrix P and a diagonal matrix D such that P-'AP = D.

A:

Q: If A and B are arbitrary real m X n matrices, then the mapping (A, B) = trace(A" B) defines an inner…

A:

Q: TRUE/FALSE Assume that all the given matrices are n × n. det((AB)T ) = det(A) det(B)

A:

Q: (i) Use the definition of inverse to prove the following: if A and B are n × n nonsingular matrices,…

A:

Q: Let A and B be n × n matrices such that AB = I. Prove that ∣A∣ ≠ 0 and ∣B∣ ≠ 0.

A: Since AB = I, therefore determinant of AB is 1, that is |AB|=1. This implies that AB is invertible.…

Q: Let X be the 3 x 3 matrix such that left-multiplication by X has the same effect as applying the…

A: In this question we have to find the (1, 2) entry of x.

Q: Let A be a 3 x 3 matrix such that det(A) = 2. Evaluate det(Aª) and det(2A).

A: Recall the following. If A is a n×n matrix and m be any positive scalardet k A=kndet Adet Am=det Am

Q: If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn…

A: Given : If A and B are square matrices of the same order such that AB = BA, then prove by induction…

Q: TRUE/FALSE Assume that all the given matrices are n × n. det(cA) = c det(A)

A:

Q: Let X = ( 0 −1 ) and Y = ( 0  1 ) be elements in SL2(R).               [ 1   0 ]              [ -1…

A:

Q: If A is an n X n matrix, prove that det(adj A) = (det A)n-1

A: Given the matrix A of order n×n, and we have to prove that det(adjA) = (detA)n-1

Q: 3. Write an example of matrices B, A such that A is k x n and B is n x k for some positive integers…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Let A and B be invertible n×n matrices. Show that det(A)=det(B) if and only if ?=??A=UB where U is a…

A: This question is related to matrix, we will solve it , we will solve it using given information.

Q: If A is a skew-symmetric n X n matrix and n is odd, prove that det A = 0.

A: Since A is a skew-symmetric matrix we have,  AT=-A where AT is the transpose of matrix A. Then we…

Q: Prove: If A and B are m × n matrices such that Ax = Bx for all x ∈ R n , then A = B.

A: Given, If A and B are m×n matrices such that Ax=Bx for all x∈Rn Prove that: A=B

Q: Prove that if A and B are similar n xn matrices, then tr(A) = tr(R). %3D

A:

Q: Let A be a 3 × 3 matrix such that Atr A = 4I3. Prove that det A = ±8. (Here, I3 is the 3 x 3…

A:

Q: Let A and B be n × n matrices such that Ax = λx andBx = μx. Show that(a) (A + B )x = (λ + μ)x;(b)…

A: We have, A and B be n x n matrices with eigenvalues λ and μ respectively, such that Ax=λx and Bx=μx.…

Q: Let A and C be n×n matrices such that ​CA=I​ (the n×n identity​ matrix). Show that Ax​=0 has only…

A: Here, CA=I, where I is the identity matrix.

Q: 4. Let M (for matrices) be the set of all 2 x 2 matrices over the reals and x, y ∈AM. Does…

A: This statement is not true.

Q: Select all statements below which are true for all invertible n X n matrices A and B A. (AB)-! = A-'…

A: In the question we have to choose right answers .

Q: Prove: If A and B are n × n diagonal matrices, then AB = B A.

A: Given, A and B are n × n diagonal matrices

Q: Show that if A and B are any m×n and n×m matrices respectively, and m > n,then ABi s not invertible.

A:

Q: Prove: For all m,n E N, (A†)™ = (A")† holds for all m x n matrices A.

A: Given that m,n∈ℕ. Let A be an m×n matrix. Let A+=Z. We know that A+=Z is pseudo inverse of A if it…

Q: (a) The set of all diagonal n xn matrices (b) The set of all n xn matrices A such that det(A) = 0.…

A: The set of all diagonal matrix.   Yes, subspace.   As, some of two diagonal matrix is a diagonal…

Q: n ×n matrices A, B, C, D where A is invertible and AC = CA, show that: det(matrix[ A B C D ]) =…

A: Given - A, B, C, D are n×n matrices where  A is invertible             and AC=CA We have to show…

Q: 27. For any matrices A and B for which the product AB is defined, the (L0-entry of AB equals a bu

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Given n >1, show that the set of all 1 x n matrices of positive integers is countable.

A: Let Z be the set of all integers. We know that Z is an countable set. We have to show that set of…

Q: Let A and B be symmetric n × n matrices. Prove that AB = BA if and only if AB is also symmetric.

A: Given,  A matrices  A and B are symmetric of order (n x n).

Q: TRUE/FALSE Assume that all the given matrices are n × n. det(A + B) = det(A) + det(B)

A: Consider the two matrices: And

Q: (a) Prove that if A and B are symmetric n X n matrices, then so is A + B. (b) Prove that if A is a…

A: Part-(a) : Given : A and B are symmetric n×n matrices. We have to prove that A + B is symmetric n×n…

Q: 16. Prove that for any square matrix A, ||A||ž E ||A* A||2.

A: Given: A is a square matrix.

Q: Find 2 x 2 matrices A and B such that det(A + B) det A + det B.

A: We have to find matrices A and B such that det(A + B) ≠ det A + det B

Q: Select all statements below which are true for all invertible n x n matrices A and B A. (ABA-¹)2 =…

A:

Q: If T,(x, y) = (2x+ y, x, x- y) and T, (x, y,z) = (x- y+z, x + y). Find the standard matrices and…

A:

Q: Prove that if A and B are matrices such that A is symmetric, then (BA−1)T(A-1BT)-1=In

A:

Q: For every real nx n matriz A, there is a s either a Householder matrix or the identity, R=H₂H₂H₁A.

A:

Q: For each of the following symmetric matrices A, find a matrix Q such that Q- = Q" and Q'AQ is…

A:

Q: Find a 3×3 matrix A such that  Ax=4x for all x in R3.

A: To find a 3×3 matrix A such that Ax=4x for all x in R³

Q: Show that for any matrices A E RmXn, BE R"XP, ||AB||F < ||A||2||B|F Hint: Write B = [b] ...bp] and…

A: It is given that A∈ℝm×n,B∈ℝn×p Let A=a1Ta2T⋮amT where ai's are m column vectors corresponding to…