Question

If A=\(\begin{bmatrix}cos⁡x&-sin⁡x&-cos⁡x\\sin⁡x&-cos⁡x&sin⁡x \end{bmatrix}\). Find A’A.

(a) \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)

(b) \(\begin{bmatrix}1&0&1\\1&0&1\\1&0&1\end{bmatrix}\)

(c) \(\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}\)

(d) \(\begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}\)

I got this question at a job interview.

Question is from Transpose of a Matrix topic in portion Matrices of Mathematics – Class 12

Answer 1

Right answer is (d) \(\begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}\)

To elaborate: Given that, A=\(\begin{bmatrix}cos⁡x&-sin⁡x&cos⁡x\\sin⁡x&cos⁡x&sin⁡x \end{bmatrix}\)

∴ A’=\(\begin{bmatrix}cos⁡x&sinx\\-sin⁡x&cos⁡x\\cos⁡x&sin⁡x \end{bmatrix}\)

⇒A’ A=\(\begin{bmatrix}cos⁡x&sin⁡x\\-sin⁡x&cos⁡x\\cos⁡x&sin⁡x \end{bmatrix}\)\(\begin{bmatrix}cos⁡ x &-sin⁡x&cos⁡x\\sin⁡x&cos⁡x &sin⁡x \end{bmatrix}\)

=\(\begin{bmatrix}cos^2⁡x+sin^2⁡x&-sin⁡x  cos⁡x+sin⁡x  cos⁡x&cos^2⁡x+sin^2⁡x\\-sin⁡x  cos⁡x+sin⁡x  cos⁡x&sin^2⁡x+cos^2⁡x&-sin⁡x  cos⁡x+cos⁡x  sin⁡x\\cos^2⁡x+sin^2 x&-cos⁡x  sin⁡x+sin⁡x cos⁡x &cos^2⁡x+sin^2⁡x \end{bmatrix}\)

=\(\begin{bmatrix}1&0&1\\0&1&0\\1&0&1\end{bmatrix}\)