Prove that, for square matrices A and B, AB = BA if and only if (A - B)(A + B) = A2 - B2

Q: Prove that if A and B are n × n orthogonal matrices, then AB and BA are orthogonal.

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Q: Show that A and B are not similar matrices

A: Given matrix is The characteristic polynomial of matrix A is

Q: Find matrices P and D so that A = PDP1 and D is a diagonal

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Q: Suppose A. B and X are 'nxn matrices such that A and B are invertible, and Which one of the…

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Q: If A and B are invertible 5x5 matrices, then A+B is also invertible. True False

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Q: .

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Q: Show that AB and BA are not equal for the given matrices.

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Q: Show that AB and BA are not equal for the matrices

A: Given: The matrices are given as, Finding the value of AB by using matrix multiplication,

Q: Show that AB and BA are not equal for the given matrices.

A:

Q: Show that for 2 × 2 matrices A & B the following equation holds. det(A + B)? = [det(A + B)]² Without…

A: a) Show that for 2x2 matrices A and B the following equations hold               det(A+B)2=det(A+B)2…

Q: For any two 3 x 3 matrices A and B, we have (А + B)(А — В) — А? — АВ + ВА - В?

A: Given statement: For any two 3×3 matrices A and B, we have

Q: If A, B, C are square matrices of the same size such that AB = A C, then for sure B = C. True False

A: See the solution below.

Q: Prove that, for square matrices A and B, AB = BA if and only if (A B)(A + B) = A2 - B2.

A: Given: for square matrices A and B, AB = BA if and only if (A B)(A + B) = A2 - B2.

Q: Show that AB and BA are not equal for the given matrices. -2 1 8 0 A = 0 6 -1 2 АВ ВА

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Q: - 1 2 A = 5 , B = -2 2 4 5 -3

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Q: If A and B are symmetric matrices of the same order, then show that ABis symmetric if and only if A…

A: Given that, A and B are symmetric matrices.

Q: Given two nxn matrices A and B, what are the criteria for the matrices to be inverses?

A: Given the matrices A and B of order nxn.

Q: Use a counterexample to show that the statement is false. If A, B, and C are matrices and AC = BC,…

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Q: If A and B are n x n matrices, and vector v is in the image of both A and B, then i must be in the…

A: True or false.

Q: 43. For square matrices A and B, is it true, in general, that (A + B)-1 = A¯1 + B¯!? Explain.

A: Explanation of the answer is as follows

Q: Find two 2 × 2 matrices A such that A2 = I.

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Q: If A and B are m × n matrices such that Col(A) and Col(B) are isomorphic, then the RREF of both…

A: If two vector space is Isomorphic then the dimension of the vector space must be same. The dimension…

Q: If AB + BA is defined, then A and B are square matrices of the same size. True False

A: Given AB + BA is defined.

Q: Find 2 × 2 matrices A and B such that AB = O but BA ≠ O.

A: To Find: 2 x 2 matrices A and B such that AB = O but BA is not equal to O Here O is a zero matrix…

Q: Find nonzero 2 × 2 matrices A and B such that AB = O.

A: Consider

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: Show that if A and B are n × n matrices and N(A − B) = Rn then A = B.

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Q: Let A and B be n × n matrices and let C = A − B. Show that if Ax0 = Bx0 and x0 = 0, then C must be…

A: Consider the A and B are n×n matrices and C=A-B. If x0=0, Ax0 =0 and Bx0 =0 are nontrivial…

Q: Show by direct calculation for 2x2 matrices A, B, C that A(BC)

A: To prove that  for 2×2 matrices A, B,  that A(BC)=(AB)C.

Q: Find the inverse of the matrix A, then prove that Ax A-1 =1, 3 -2 1) A =

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Q: 1. For any 3x3 matrices A & B: Verify that (AB)" = B"A". %3D

A: By using the properties

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

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Q: Let A and B be square matrices. Show that even though AB and BA may not be equal, it is always true…

A: Given : A and B are square matrix

Q: Show that if a matrix A has an inverse A-, then in fact A-1 has an inverse as well

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Q: If A and B are similar, then there exists a matrix P such that BK = P-AKP. Use this fact to find B,…

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Q: Give an example to show that if A and B are symmetric n X n matrices, then AB need not be symmetric

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Q: Which of the following statements is true for all invertible 3x3 matrices A and B? (3ABA-!)-|B =…

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Q: If A and B are nXn matrices, then (AB)²=A²B? A True

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Q: If A is a square matrix, then A2 is defined. TRUE OR FALSE

A: For matrix multiplication to be defined, the number of columns in the first matrix must be equal to…

Q: Find a pair of unequal 2 × 2 matrices A and B , other than those given in Example 9, such that AB =…

A: Let A=0-10  2  &  B=3500.

Q: If A, B, and C are square matrices of the same order such that AC = BC, then A = B. True False

A: Given AC = BC

Q: Prove the following: A(BC) = (AB)C for all 2x2 matrixes

A: Consider the matrices: A=a1a2a3a4B=b1b2b3b4C=c1c2c3c4

Q: Create two 2 x 2 matrices A and B, to show that (A+ B)- A +B is not always true. Clearly write A, B,…

A: Give an example of 2 × 2 matrix, in which A + B-1 = A-1 + B-1 is wrong.

Q: . We can add (or subtract) two matrices only if they have thesame .

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Q: Show that AB and 8A are not equal for the given matrices. 11 5 5 AB BA 151/2 1515 1會

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Q: Show that for any square matrices A and B, Tr (AB) = Tr (BA)

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Q: Find two unequal 2 × 2 matrices A and B such that

A: The objective is to find two different 2 × 2 matrices A and B such that AB=1001 Let A=pqrs, B=tuvw…

Q: Show by counterexample that (AB)−1 ≠ A−1B−1. That is, find two matrices A and B for which (AB) -1 ≠…

A: Let

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.