22 Let A, B E Mnxn (F). We say that A is similar to B, denoted by A B, if there exists an invertible matrix Q € Mnxn(F) such that A = Q-¹BQ. Prove the following statements. (a) A~ A for any A € Mnxn (F). (b) If A, B E Mnxn (F) and A ~B, then B~ A. (c) If A, B, C = Mnxn (F) and A ~ B and B~ C, then A~ C.

Q: Let A be an invertible matrix. Show that det (adj (A)) = det (A-¹).

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Q: 7. Let A be an invertible matrix. Prove that if A-¹ = AT, then det(A) = +1.

A: Given: A is an invertible matrix. If A-1=AT, to prove that detA=±1

Q: Prove that (a) C(B) C Nul(A), where C(B) denotes the column space of B. (b) rank A + rank B < 5

A: Given that A is a 3×5 matrix and B is a 5×6 matrix. Also AB=0.

Q: Let A be a matrix whose entries are integers, and det A=1. 4 : Show that the entries of A-1 are all…

A: given ,det A = 1

Q: 65. Final exam, Winter 2015. A matrix A is said to be skew-symmetric if AT = -A. For instance 03 М-…

A: 65.(a) A is skew symmetric matrix then AT=-A Using the results: (1) det(AT)=det(A) (2) det(cA)=cn…

Q: Q9. As usual, let In denote the n x n identity matrix. 1 - (a) Find a 2 × 2 matrix such that A² =…

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Q: Show that the matrices [1 1 1] A = |1 3 0 and B = |0 o 0 0 1 1 1 1 are similar.

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Q: Suppose A is a symmetric matrix. (Au)" : u"A

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Q: 7. Let A and B two 3-by-3 matrices, with |B| = 1. Let C be the inverse matrix of B. Which of the…

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Q: Exercise 2. Suppose that A and B are matrices of the same size. Prove the following: (a) (aA)¹ = aAª…

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Q: lar, let m = 2 and n e of A and B such that

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Q: 34. Is it possible for A³ to be an identity matrix without A being invertible? Explain.

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Q: -1 [A] = 4 3 -3 2 %3D L3 1] 2.

A: Given :                    2-134-32321 To find :                 The cofactors for C13 , C22 , C32…

Q: For each matrix A with real valued entries, find a diagonal matrix D and an invertible matrix Q such…

A: Part(a): Given matrix is, A=1332 Now, finding the eigenvalues of the given matrix,…