If $A=\begin{bmatrix}x&y&z\\y&z&x\\z&x&y\end{bmatrix}$

where $x, y, z$ are integers, is an orthogonal matrix, then what is $A^{2}$ equal to?

Question

Question

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Given,

Matrix A is defined as:

$A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$

It is mentioned that A is an orthogonal matrix. Therefore,

$A^T A = I$

Now, we calculate A²:

$A^2 = A \cdot A$

$A^2 = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix} \cdot \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$

Since A is orthogonal, $A^T A = I$, and hence:

$A^2 = I$

∴ A² = Identity matrix (I).

Hence, the Correct answer is Option 2.

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