If \( A=\begin{bmatrix}1&0&0\\0&1&0\\2&3&-1\end{bmatrix} \), then \( A^{2} \) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → a unit matrix of order 3
Row i of \( A^{2} \) is (row i of A) multiplied by A, that is, a combination of the rows of A with the entries of row i as multipliers.
Rows 1 and 2 of A are (1, 0, 0) and (0, 1, 0), so they pick out rows 1 and 2 of A unchanged: (1, 0, 0) and (0, 1, 0).
Row 3 of A is (2, 3, −1), so row 3 of \( A^{2} \) is
\( 2(1,0,0)+3(0,1,0)-(2,3,-1) \)
\( =(0,0,1) \)
So \( A^{2}=I \), the unit matrix of order 3.