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Study the Assertion and Reason and choose the correct option.
Let the matrices A = [-3 2; -5 4] and B = [4 -2; 5 -3] be such that A¹⁰⁰B = BA¹⁰⁰.
Assertion: Let the matrices $A = \begin{bmatrix} -3 & 2 \\ -5 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -2 \\ 5 & -3 \end{bmatrix}$ be such that $A^{100}B = BA^{100}$.
Reason: $AB = BA$ implies $A^n B = BA^n$ for all positive integers $n$.
- (a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
- (b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
- (c)Assertion is true and Reason is false.
- (d)Assertion is false and Reason is true.
Answer
Answer
Correct option: (a)
Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
$AB = \begin{bmatrix}-12+10 & 6-6\\ -20+20 & 10-12\end{bmatrix} = \begin{bmatrix}-2&0\\0&-2\end{bmatrix}$ and $BA = \begin{bmatrix}-12+10 & 8-8\\ -15+15 & 10-12\end{bmatrix} = \begin{bmatrix}-2&0\\0&-2\end{bmatrix}$, so $AB = BA$.
The Reason is true (repeatedly commute $B$ past $A$), so with $n = 100$, $A^{100}B = BA^{100}$: the Assertion is true and follows from the Reason.