If $A=\begin{bmatrix} 4 & 5 & 2\\3 & -1 & 7 \end{bmatrix},$ then the sum of the elements of the matrix $AA^T$ is :

146

The correct answer is Option (3) → 146

$A=\begin{bmatrix} 4 & 5 & 2\\3 & -1 & 7 \end{bmatrix}_{2×3},A^T=\begin{bmatrix}4&3\\5&-1\\2&7\end{bmatrix}_{3×2}$

$∴AA^T=\begin{bmatrix}16+25+4&12-5+14\\12-5+14&9+1+49\end{bmatrix}$

$=\begin{bmatrix}45&21\\21&59\end{bmatrix}$

$∴45+59+21+21=146$