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Matrics — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMatrics2 Marks
MCQ
If $A=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$ such that $A^2-4 A+3 I=0$ where $I$ is a unit matrix of order, 2 , then $A^{-1}$ is
  • A
    $\frac{1}{3}\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$
  • B
    $\frac{1}{3}\left[\begin{array}{cc}-1 & 2 \\ 2 & -1\end{array}\right]$
  • C
    $\frac{1}{3}\left[\begin{array}{cc}-2 & 1 \\ 1 & -2\end{array}\right]$
  • ✓
    $\frac{1}{3}\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
✓

Answer

Correct option: D.
$\frac{1}{3}\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
(d) : Given, $A^2-4 A+3 I=0$
Postmultiply by $A^{-1}$ both sides,
$
\begin{aligned}
& A-4 I+3 A^{-1}=0 \\
& \text { or } 3 A^{-1}=4 I-A \\
& A^{-1}=\frac{1}{3}(4 I-A)=\frac{1}{3}\left[\left[\begin{array}{ll}
4 & 0 \\
0 & 4
\end{array}\right]-\left[\begin{array}{cc}
2 & -1 \\
-1 & 2
\end{array}\right]\right]=\frac{1}{3}\left[\begin{array}{ll}
2 & 1 \\
1 & 2
\end{array}\right]
\end{aligned}
$

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