Which of the following matrices are invertible? $\mathrm{A}=\left[\begin{array}{cc} 2 & 3 \\ 10 & 15…

Which of the following matrices are invertible? $\mathrm{A}=\left[\begin{array}{cc} 2 & 3 \\ 10 & 15 \end{array}\right], \mathrm{B}=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3 \end{array}\right], \mathrm{C}=\left[\begin{array}{lll} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{array}\right], \mathrm{D}=\left[\begin{array}{lll} 2 & 4 & 2 \\ 1 & 1 & 0 \\ 1 & 4 & 5 \end{array}\right]$
  1. both A and B
  2. only C
  3. only A
  4. only D

Solution

$\begin{aligned} & |A|=\left[\begin{array}{cc}2 & 3 \\ 10 & 15\end{array}\right]=30-30=0 \mid \\ & |B|=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3\end{array}\right]=0 \quad \ldots\left[\mathrm{R}_1 \text { and } \mathrm{R}_2 \text { are identical }\right] \\ & |\mathrm{C}|=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8\end{array}\right]=1(2)-2(4)+3(2)=0 \\ & |\mathrm{D}|=\left[\begin{array}{lll}2 & 4 & 2 \\ 1 & 1 & 0 \\ 1 & 4 & 5\end{array}\right]=2(5)-4(5)+2(3)=-4 \neq 0\end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 1)

Practice more Matrices questions on Aicharya