Problem (i) Let A and B be symmetric nxn matrices over R. Show that AB is symmetric if and only if A and B commute. (ii) Let A, B be n x n hermitian matrices. Is i[A, B] hermitian? (iii) Let A, B be symmetric n xn matrices over R. Show that [A, B] is skew- symmetric over R.

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Q: Determine all 2 × 2 matrices A such that AB = B A for any 2 × 2 matrix B.

A: Let A=abcd and B=pqrs be any 2×2 matrices.

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

Q: Let A and B be two square matrices of the same size. Prove that det(AB) = det(BA).

A: Using Determinant and elementary row operations,

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Q: 8. (3pts) Let A and B be arbitrary n x n matrices. Multiply (A – B)(A+ B).

A: As we know that (A-B)(A+B) = AA+AB-BA-BB=A2+AB-BA-B2here AB need not to be equal to the BA. Because…

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Q: -() --()-() 3 6 3 3 1 2 1 2 1 1 2 1 3 1 2 3 2 2 A = 3 1 2 В C = 4 4 3 4 4 3

A: Claim: We have to find E such that EA=Bi.e. E=BA-1 And F such that, FA=C i.e. F=CA-1 A=121312443then…

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A: To state true or false given statement.

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A: Given: It is given that A,B=AB-BA where, A and B are square matrices. To prove: We have to prove…

Q: Let A, B, C be invertible matrices of order 3x3. Show that if AB = AC , then B =C. %3D

A: Let A, B, C be invertible matrices of order 3×3. To show that if AB=AC, then B=C.

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A: The given statement is false.

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A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

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