First slide

Operations of matrix

Question

If AB = A and BA = B, where A c. · B are square matrices, then

Moderate

A

$B^{2} = B$ and $A^{2} = A$
B

$B^{2} \neq B$ and $A^{2} = A$
C

$A^{2} \neq A, B^{2} = B$
D

$A^{2} \neq A, B^{2} \neq B$

Solution

We have,
$A^{2} = A A$
$\begin{array}{l} \Rightarrow A^{2} = (A B) A [∵ A B = A] \\ \Rightarrow A^{2} = A (B A) \\ \Rightarrow A^{2} = A B [∵ B A = B] \\ \Rightarrow A^{2} = A [∵ A B = A] \end{array}$
and,
$\begin{array}{l} B^{2} = B B \\ \Rightarrow B^{2} = (B A) B [∵ B A = B] \\ \Rightarrow B^{2} = B (A B) \\ \Rightarrow B^{2} = B A [∵ A B = A] \\ \Rightarrow B^{2} = B [∵ B A = B] \end{array}$

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