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If A = \( \begin{bmatrix} 1& -1 \\[0.3em] 2& -1 \\[0.3em] \end{bmatrix}\) , B = \( \begin{bmatrix} a& -1 \\[0.3em] b& -1 \\[0.3em] \end{bmatrix}\) and (A + B)2 = (A2 + B2) then find the values of a and b.

A = [(1,-1)(2,-1)]

B = [(a,-1)(b,-1)]

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Given : A = \( \begin{bmatrix} 1& -1 \\[0.3em] 2 & -1 \\[0.3em] \end{bmatrix}\) , B = \( \begin{bmatrix} a& -1 \\[0.3em] b & -1 \\[0.3em] \end{bmatrix}\) 

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

To find : a and b

Formula used :

Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj

If A is a matrix of order a x b and B is a matrix of order c x d ,then matrix AB exists and is of order a x d ,

if and only if b = c

It is given that (A + B)2 = (A2 + B2)

Equating similar terms in the given matrices we get,

2 – 2a = -a + 1 and -2b = -b + 1 2 – 1 

= -a + 2a and -2b + b = 1 

1 = a and -b = 1 

a = 1 and b = -1

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