Q112 / 170 · Manipal (MET) 2021 · Mathematics

If $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]$, then $A^2$ is equal to

  • Correct answer
    A
    unit matrix
  • B
    null matrix
  • C
    $A$
  • D
    $-A$

Solution

Given, $A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]$ $\begin{aligned} \therefore A^2 & =A \cdot A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right] \\ & =\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a-a & b-b & 1\end{array}\right]=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\end{aligned}$

Hence, the correct answer is option: 'A'.

Relevant Questions