Let A and B be two 3×3 matrices such that AB = BA. Further, if A≠B4 and A2=B8 , then which of the following is/are correct?

| A + B 4 | ≠ 0 , There exist a 3 × 3 matrix C such that ( A + B 4 ) C is null matrix , | A + B 4 | = 0

Let A and B be two 3×3 matrices such that AB = BA. Further, if A≠B4 and A2=B8 , then which of the following is/are correct?

There exist a non-zero 3 × 3 matrix C such that ( A + B 4 ) C is null matrix

There exist a 3 × 3 matrix C such that ( A + B 4 ) C is null matrix

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Let A and B be two $3 \times 3$ matrices such that AB = BA. Further, if $A \neq B^{4}$ and $A^{2} = B^{8}$ , then which of the following is/are correct?

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$| A + B^{4} | \neq 0$

There exist a non-zero $3 \times 3$ matrix C such that $(A + B^{4}) C$ is null matrix

There exist a $3 \times 3$ matrix C such that $(A + B^{4}) C$ is null matrix

$| A + B^{4} | = 0$

answer is A, C, D.

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Detailed Solution

$(A - B^{4}) (A + B^{4}) = 0... (1) ∵ A B = B A$
$\Rightarrow | A + B^{4} | | A + B^{4} | = 0$
Now if $| A + B^{4} | \neq 0 \Rightarrow A_B^{4} = 0 by (1)$
Which is not possible $∴ | A + B^{4} | = 0$
Which is like matric C for which $(A + B^{4}) C = 0$ so A, C, D are correct