ISI2016-DCG-31

Question

Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$  stands for determinant of matrix $A.$ Then

• A. $\mid\:(A)\mid=1$
• B. $\mid\:(A)\mid=0\:\text{or}\:1$
• C. $\mid\:(A)\mid=-1,0\:\text{or}\:1$
• D. $\mid\:(A)\mid=-1\:\text{or}\:1$

Answer 1

$\mid A^{2} \mid = 1$

$\implies \mid A \mid ^{2}= 1\rightarrow (1)$

Let $\mid A \mid  = x,$ put the value in equation$(1),$ then we get

$x^{2} = 1\implies x = \pm 1$

Therefore $,\mid A \mid = \pm 1\implies \mid A \mid  = -1\:\: \text{or}\:\: 1$

So, the correct answer is $(D)$.

Reference:

• https://math.stackexchange.com/questions/900897/how-matrix-a-is-called-a2-i

Comments

It will be option D..right?

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Yes