If $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$, then $A A^T$ is a

If $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$, then $A A^T$ is a
  1. symmetric matrix
  2. skew-symmetric matrix
  3. singular matrix
  4. inverse of $A$

Solution

Here, $\begin{aligned} A & =\left[\begin{array}{ccc}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right] \\ A^T & =\left[\begin{array}{ccc}3 & 2 & 0 \\ -3 & -3 & -1 \\ 4 & 4 & 1\end{array}\right]\end{aligned}$ $\begin{aligned} A A^T & =\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]\left[\begin{array}{ccc}3 & 2 & 0 \\ -3 & -3 & -1 \\ 4 & 4 & 1\end{array}\right] \\ A A^T & =\left[\begin{array}{lll}9+9+16 & 6+9+16 & 0+3+4 \\ 6+9+16 & 4+9+16 & 0+3+4 \\ 0+3+4 & 0+3+4 & 0+1+1\end{array}\right]\end{aligned}$ $\begin{aligned} & A A^T=\left[\begin{array}{lll}34 & 31 & 7 \\ 31 & 29 & 7 \\ 7 & 7 & 2\end{array}\right] \\ & \left(A A^T\right)^T=\left[\begin{array}{ccc}34 & 31 & 7 \\ 31 & 29 & 7 \\ 7 & 7 & 2\end{array}\right]\end{aligned}$ So, $A A^T$ is a symmetric matrix.

Asked in: AP EAMCET 2022 (05 Jul Shift 1)

Practice more Matrices questions on Aicharya