First slide

Operations of matrix

Question

$Let A, B be two n \times n matrices such that A + B = A B, then$

Moderate

A

$A B = B A$
B

${AB = I}_{n}$
C

$A = I_{n} or B = I_{n}$
D

None of these

Solution

$Step-I: (A - I) (B - I)$
$= A B - A - B + I$
$= A B - (A + B) + I$
$= A B - A B + I (∵ A + B = A B) (\begin{array}{l} A B = I \\ B A = I \end{array})$
$Step-II: (B - I) (A - I) = I$
$B A - B - A + I = I$
$B A - (A + B) = 0 (∵ A + B = A B)$
$B A - A B = 0$
$A B = B A$

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