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Q.

If $A$ and $B$ are two matrices such that $A + B = A B,$ then

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a

$A B = I$

b

$A B = B A$

c

$A = I$

d

$B = I$

answer is C.

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Detailed Solution

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$\begin{array}{l} A + B = A B \\ \Rightarrow I - (A + B) + A B = I \\ \Rightarrow (I - A) (I - B) = I \\ \Rightarrow (I - A)^{- 1} = I - B \\ \Rightarrow (I - B) (I - A) = I \\ \Rightarrow I - (B + A) + B A = I \\ \Rightarrow A + B = B A \end{array}$
Thus, $A B = B A$

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If A and B are two matrices such that A+B=AB, then