Let A be a square matrix such that AAT = I. Then 1/2 A[(A + A^T)^2 + (A − A^T)^2] is equal to

Question

Let \(A\) be a square matrix such that \(A A^{T}=I\). Then \(\frac{1}{2} A\left[\left(A+A^{T}\right)^{2}+\left(A-A^{T}\right)^{2}\right]\) is equal to

(1) \(A^{2}+I\)

(2) \(A^{3}+I\)

(3) \(A^{2}+A^{T}\)

(4) \(A^{3}+A^{T}\)

Answer 1

Correct option is (4) \(A^{3}+A^{T}\)

\(\mathrm{AA}^{\mathrm{T}}=\mathrm{I}=\mathrm{A}^{\mathrm{T}} \mathrm{A}\)

On solving given expression, we get

\(\frac{1}{2} \mathrm{~A}\left[\mathrm{~A}^{2}+\left(\mathrm{A}^{\mathrm{T}}\right)^{2}+2 \mathrm{AA}+\mathrm{A}^{2}+\left(\mathrm{A}^{\mathrm{T}}\right)^{2}-2 \mathrm{AA} \mathrm{A}^{\mathrm{T}}\right]\)

\(=\mathrm{A}\left[\mathrm{A}^{2}+\left(\mathrm{A}^{\mathrm{T}}\right)^{2}\right]\)

\(=\mathrm{A}^{3}+\mathrm{A}^{\mathrm{T}}\)