Q31 of 146 Page 52

Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.

By matrix multiplication we can write:


(A + B)2 = (A+B)(A+B) = A2 + AB + BA + B2


We know that matrix multiplication is not commutative but it is given that : AB = BA


∴ (A + B)2 = A2 + AB + AB + B2


⇒ (A + B)2 = A2 + 2AB + B2 …proved


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29

Show that A’A and AA’ are both symmetric matrices for any matrix A.

 30

Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2 ? Give reasons.

 32

Let and a = 4, b = –2.

Show that:


A + (B + C) = (A + B) + C

32

Let and a = 4, b = –2.

Show that:


A(BC) = (AB)C