4. Let M (for matrices) be the set of all 2 x 2 matrices over the reals and x, y ∈AM. Does commutativity ,that is, xy = yx, hold true for multiplication? Prove or disprove.

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

A:

Q: Prove or disprove: For any n×n matrix A, AT A = AAT

A: For any n×n matrix A, ATA=AAT

Q: Suppose A and B are n×n matrices such that B is invertible and AB is invertible. Prove that A is…

A: Given that the matrices A and B are of order n×n and both A, AB are invertible.

Q: 4. Consider the matrices U = V = 2 -3 Calculate the following, where legal: (a) 4U (b) UUT (c) UV

A: Here the given matrices are                                             U=110-2   and V=012-311…

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

A: Given: Let A and B be n ×n matrices.

Q: 2. Compute (3C – 4D)², where C and D are arbitrary n x n matrices. Hint: Remember that matrix…

A: To compute, 3C-4D2

Q: Verify that det(AB) = (det A)(det B) for the following matrices: 3 0 4 5 2 3 6 A = 0 3 0 0 2 B = -4…

A:

Q: Suppose A, B and X are matrices such that AX A = B, where 0 1 0 1 -2 1 3. -134 -5 1 2 , then X = |…

A:

Q: 3. Let A be a m X n matrix with columns C1, ... , Cn . Prove that {c1, ..., C,} is independent if…

A: In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a…

Q: Let A and B be n × n matrices. Prove that if A is nonsingular, then AB is similar to BA.

A: It is given that matrices  A and B are of order (n x n). To show that AB is similar to BA. Since,…

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 A and B be 5x5 matrices such that det(A) = 4 and det(B) = 3. %3D Compute det(2B)"(AB)")

A: We have to find

Q: A and B are n X n matrices If Bis invertible, prove that det(B-1AB) = det(A)

A: A and B are n X n matrices and B is invertible.

Q: Let K be an nxn matrix over Z26 such that K = K-1. Explain why either det(K) = 1 or %3D det (K) =…

A:

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

A:

Q: Suppose that A, B are n x n matrices. Prove that if A is singular, then B is singular.

A:

Q: TRUE/FALSE Assume that all the given matrices are n × n. If A = O, but Ak = O (where O denotes the…

A: Given:   To Check: A is singular or not Concept: A is singular only when det(A) = 0

Q: Prove that if A and Bare sk ew-symmetric n X n matrices, then so is A + B

A: Given: To Show:   Concept:

Q: 4. Prove that det (A) = det (A¹) for a 2 × 2 matrix.

A:

Q: 8. Prove or give a counterexample. Assume all relevant matrices are square and of the same size. (a)…

A:

Q: 13. Find all 2 x 2 matrices A = such that A? %3D

A:

Q: / Let H, K are two zero-one matrices H= K= 1) Join H and K 2) Is H symmetric? Why? 3) НОК 4) K³

A:

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

A:

Q: Let A be a m × n matrix with columns c1, …, cn. Prove that {c1, …, cn} is independent if and only if…

A:

Q: Let A and B be n × n matrices such that AB is singular. Prove that either A or B is singular.

A: Let A and B be n × n matrices such that AB is singular.

Q: Let Q be a orthogonal matrix prove that If Q1 and Q2 are orthogonal n X n matrices, then so is Q1…

A:

Q: 2. If in the general state(x, Ax) = (x, Bx be for each x at Euclidean space, then B = A is not…

A: A matrix is symmetric when the matrix equals to its transpose. When a matrix equals negative of its…

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: ☺ Let M(for matrices> be the set over the reals and Xiy of all 2x2 motrices EM .Does | , thst is…

A:

Q: Let A and B be two square matrices of order b x n. Prove that A and B are non singular if and only…

A:

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

A: If we have :                     A as an n×n matrix;   To prove:                            If Ax=x…

Q: Let A be an n × n matrix. Prove that if Ax = 0 for all n × 1 matrices x, then A = O .

A: we have been given that Ax = 0 for all n × 1 matrix x and  A be an n × n matrix.

Q: 4. Suppose B, I – B-' are invertible matrices and (X – A)- = BX-1, find X (as an expression in A,…

A: A matrix Y is said to be invertible if and only if Y-1 Y=Y Y-1=I

Q: 2. Verify, without using the matrix for 0(), that [O()]° is the identity matrix 1 0 0 1

A: This is a problem of rotation of the matrix.

Q: Show that if A is invertible, then adj A is invertible, and (adj A)~ %3D det A [Hint: Given matrices…

A: If we can show the following then we can easily show that adj A is invertible. By using the hint:

Q: 13. Find all 2 x 2 matrices A = (" ta² = (C :) such that %3D

A: We need to find the set of all 2×2 matrices such that A2=O, where O is the zero matrix of order 2.

Q: 6. Suppose that A, B, C are n x n matrices and ABC = I. Find A- in terms of A,C 7. For any non-zero…

A: (A) To find A-1 in terms of A, C if ABC = I and A, B, C are nxn matrices. (B) To determine whether…

Q: Let A and B be equivalent square matrices. Prove that A is nonsingular if and only if B is…

A: Given, A and B are equivalent square matrices. Let P be any non singular matrix of the same…

Q: Let A = [kaij]nxn, B = [aij]nxn, be an nxn matrices and k be a scalar then det(A) is equal to: *…

A:

Q: Let A and B be n × n matrices and let C = AB. Prove that if B is singular then C must be singular.

A:

Q: 4. Prove: If A and B are both n x n matrices, then det(AB)= detAdetB. (Only use 2x2 matrices)

A: Given A and B  n×n  matrices to prove  det(AB)=det A det B

Q: For all square matrices A and B, it is true that det(A + B) = det(A) + det(B) O True O False

A:

Q: 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 ·…

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Q: For every square matrix A and every scalar c, we have det(cA) = c det(A). O True O False

A: Given statement is  For every square matrix A and every scalar c , we have det ( cA ) = c det ( A )…

Q: 6. Show that (AA)* = X* A* for all A E C and all matrices A.

A:

Q: Is it true or false? |A+ B" |=| A" + BT for any nxn matrices A and B. Birini seçin: O Doğru O Yanlış

A: To solve the above question we must know the following determinant property of any square matrix…

Q: Let A and B be n×n matrices. Prove that the productAB is nonsingular if and only if A and B are…

A: Definition:  A matrix is said to be non-singular if its determinant value is non-zero. It is given…

Q: 13. Find all 2 x 2 matrices A = (" such that A? %3D

A: Image is attached with detailed solution.

Q: Is it true or false? |A+ B" |=| AT + B[T for any nxn matrices A and B. Birini seçin:

A: We are asked to find the statement is true or false.