Q113 / 180 · KCET 2016 · Mathematics
If \( A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right] \) then \( A^{2}-5 A \) is equal to
- AI
- B\( -I \)
- C\( 7 I \)
- Correct answerD$-7 I$
Solution
Given that, \( A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right] \)
So, \( A^{2}=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right] \)
\( =\left[\begin{array}{cc}9-1 & 3+2 \\ -3-2 & -1+4\end{array}\right]=\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right] \)
And
\( 5 A=5\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]=\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right] \)
Therefore,
\[ \begin{array}{l} A^{2}-5 A=\left[\begin{array}{cc} 8 & 5 \\ -5 & 3 \end{array}\right]-\left[\begin{array}{cc} 15 & 5 \\ -5 & 10 \end{array}\right] \\ =\left[\begin{array}{cc} -7 & 0 \\ 0 & -7 \end{array}\right]=-7 I \end{array} \]
So, \( A^{2}=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right] \)
\( =\left[\begin{array}{cc}9-1 & 3+2 \\ -3-2 & -1+4\end{array}\right]=\left[\begin{array}{cc}8 & 5 \\ -5 & 3\end{array}\right] \)
And
\( 5 A=5\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]=\left[\begin{array}{cc}15 & 5 \\ -5 & 10\end{array}\right] \)
Therefore,
\[ \begin{array}{l} A^{2}-5 A=\left[\begin{array}{cc} 8 & 5 \\ -5 & 3 \end{array}\right]-\left[\begin{array}{cc} 15 & 5 \\ -5 & 10 \end{array}\right] \\ =\left[\begin{array}{cc} -7 & 0 \\ 0 & -7 \end{array}\right]=-7 I \end{array} \]
Hence, the correct answer is option: 'D'.