Matrices & Determinants
3×3 matrices with binary entries and unit determinant
MJAT_TS7_P1
Grade 12
Question:
Let $S=\left\{A=\begin{pmatrix}a&b\\c&d\\0&1\end{pmatrix}: a,b,c,d,e\in\{0,1\}$ and $\det(A)\in\{-1,1\}\right\}$. Then the number of elements in $S$ is:
Step-by-Step Solution
Key Concept: The $3\times 3$ matrix has entries in $\{0,1\}$ with specific structure. $\det(A)=\pm 1$ means the matrix is invertible with determinant $\pm 1$. Count all such matrices.
$|S|=\mathbf{16}$.
Correct Answer: 16