Precalculus / Unit 8: Systems and Matrices
8.3 Determinants
Notes
The determinant is a number computed from a square matrix.
- For a $2 \times 2$ matrix: $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$.
- For a $3 \times 3$ matrix, expand along a row using cofactors with alternating signs $+ - +$.
- For a triangular matrix, the determinant is the product of the diagonal entries.
- A square matrix has an inverse exactly when its determinant is not 0. A matrix with determinant 0 is singular.
Practice
What is $\det\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$?
$3(4) - 2(1) = 10$.
What is $\det\begin{bmatrix} 2 & 1 & 5 \\ 0 & 3 & 7 \\ 0 & 0 & 4 \end{bmatrix}$?
The matrix is triangular, so multiply the diagonal: $2 \cdot 3 \cdot 4$.
Which matrix has no inverse?
$2(6) - 4(3) = 0$, so the matrix is singular.
Ready for more? Take the Unit 8 test.