MHT CET · Maths · Matrices
Which of the following matrices are invertible?
\(A =\left[\begin{array}{cc}2 & 3 \\ 10 & 15\end{array}\right], B =\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3\end{array}\right], C\) \(=\left[\begin{array}{ccc}1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8\end{array}\right], D =\left[\begin{array}{ccc}2 & 4 & 2 \\ 1 & 1 & 0 \\ 1 & 4 & 5\end{array}\right]\)
- A both A and B
- B only C
- C only A
- D only D
Answer & Solution
Correct Answer
(D) only D
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & |A|=\left[\begin{array}{cc}2 & 3 \\ 10 & 15\end{array}\right]=30-30=0 \mid \\ & |B|=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3\end{array}\right]=0 \quad \ldots\left[\mathrm{R}_1 \text { and } \mathrm{R}_2 \text { are identical}\right] \\ & |\mathrm{C}|=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8\end{array}\right]=1(2)-2(4)+3(2)=0 \\ & |\mathrm{D}|=\left[\begin{array}{lll}2 & 4 & 2 \\ 1 & 1 & 0 \\ 1 & 4 & 5\end{array}\right]=2(5)-4(5)+2(3)=-4 \neq 0\end{aligned}\)
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