Let A is a four order diagonal matrix whose entries are real or complex numbers such that A4=I4. If trace of A is zero then which is/are correct? (here I4 is four order unit matrix)

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Let $A$ is a four order diagonal matrix whose entries are real or complex numbers such that $A^{4} = I_{4} .$ If trace of $A$ is zero then which is/are correct? (here $I_{4}$ is four order unit matrix)

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There will be 24 distinct matrices $A$

Determinant value of all such matrices is 1

There will be 36 distinct matrices $A$

Determinant value of all such matrices is – 1

answer is B.

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Detailed Solution

$A_{4 \times 4} = (\begin{matrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d \end{matrix}) & trace of A = 0 A^{4} = (\begin{matrix} a^{4} & 0 & 0 & 0 \\ 0 & b^{4} & 0 & 0 \\ 0 & 0 & c^{4} & 0 \\ 0 & 0 & 0 & d^{4} \end{matrix}) = I & a + b + c + d = 0 ∴ a^{4} = 1 & b^{4} = 1 & c^{4} = 1 & d^{4} = 1 & a + b + c + d = 0 a, b, c, d \in {1, - 1, i, - i} a + b + c + d = 0 a, b, c, d (i) U s e a l l 1, - i, i, - i \Rightarrow 4! = 24 (i i) Use all 1, - 1, 1, - 1 \Rightarrow \frac{4!}{2! 2!} = 6 (i i i) Use all i, - i, i, - i \Rightarrow \frac{4!}{2! 2!} = 6 R . N . W = 36$