MHT CET · Maths · Matrices
If \(\mathrm{A}\) is non-singular matrix of order 3 such that \((A-2 \mathrm{I})(A-4 \mathrm{I})=0\), then \(\frac{1}{6} A+\frac{4}{3} A^{-1}\) is (where I is a unit matrix of order 3 and 0 is a null matrix of order 3)
- A 6I
- B I
- C 0
- D 2I
Answer & Solution
Correct Answer
(B) I
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & (A-2 \mathrm{I})(A-4 \mathrm{I})=0 \\ & \Rightarrow A^2-6 A+8 \mathrm{I}=0 \\ & \Rightarrow A^2+8 \mathrm{I}=6 A \\ & \Rightarrow A^2 A^{-1}+8 \mathrm{I} A^{-1}=6 A A^{-1} \\ & \Rightarrow A+8 A^{-1}=6 \mathrm{I} \\ & \Rightarrow \frac{1}{6} A+\frac{4}{3} A^{-1}=\mathrm{I}\end{aligned}\)
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