If \(A = \begin{bmatrix} 3 & 1\\ -1 & 2\end{bmatrix}\) then \(A^2 - 5A =\)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −7I
Multiplying row by column:
\( A^2 = \begin{bmatrix} 3 & 1\\ -1 & 2\end{bmatrix}\begin{bmatrix} 3 & 1\\ -1 & 2\end{bmatrix} \)
\( = \begin{bmatrix} 9 - 1 & 3 + 2\\ -3 - 2 & -1 + 4\end{bmatrix} = \begin{bmatrix} 8 & 5\\ -5 & 3\end{bmatrix} \)
\( 5A = \begin{bmatrix} 15 & 5\\ -5 & 10\end{bmatrix} \)
Subtracting:
\( A^2 - 5A = \begin{bmatrix} 8 - 15 & 5 - 5\\ -5 + 5 & 3 - 10\end{bmatrix} \)
\( = \begin{bmatrix} -7 & 0\\ 0 & -7\end{bmatrix} = -7I \)
Shortcut: For a 2 × 2 matrix the Cayley-Hamilton theorem gives \(A^2 - (\mathrm{tr}\,A)A + |A|I = O\). Here \(\mathrm{tr}\,A = 3 + 2 = 5\) and \(|A| = 6 + 1 = 7\), so \(A^2 - 5A = -7I\) at once.
NCERT (2026-27 edition):
Matrices
