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Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices2 Marks
Question
If $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]$, show that $A-4 A+3 I=0$.
✓

Answer

$\begin{aligned} & A^2-4 A+3 I \\ & =A \cdot A-4 A+3 I \\ & =\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]-4\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]+3\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ & =\left[\begin{array}{cc}4+1 & -2-2 \\ -2-2 & 1+4\end{array}\right]-\left[\begin{array}{cc}8 & -4 \\ -4 & 8\end{array}\right]+\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] \\ & =\left[\begin{array}{cc}5 & -4 \\ -4 & 5\end{array}\right]-\left[\begin{array}{cc}8 & -4 \\ -4 & 8\end{array}\right]+\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right] \\ & =\left[\begin{array}{cc}5-8+3 & -4+4+0 \\ -4+4+0 & 5-8+3\end{array}\right]\end{aligned}$

$\begin{aligned} & =\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] \\ & =0\end{aligned}$

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