Determinant Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Evaluate det⁡[2130−12104]\det\begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & 4 \end{bmatrix} by expanding along the first row.

Solution

  1. 1
    Step 1: Expand: 2det⁡[−1204]−1det⁡[0214]+3det⁡[0−110]2 \det\begin{bmatrix} -1 & 2 \\ 0 & 4 \end{bmatrix} - 1 \det\begin{bmatrix} 0 & 2 \\ 1 & 4 \end{bmatrix} + 3 \det\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}.
  2. 2
    Step 2: 2(−4−0)−1(0−2)+3(0−(−1))=2(−4)−1(−2)+3(1)2(-4-0) - 1(0-2) + 3(0-(-1)) = 2(-4) - 1(-2) + 3(1).
  3. 3
    Step 3: =−8+2+3=−3= -8 + 2 + 3 = -3.
  4. 4
    Check: The alternating signs are +,−,++, -, + for the first row ✓

Answer

−3-3
For a 3×33 \times 3 matrix, expand along any row or column using cofactors. Each cofactor is a 2×22 \times 2 determinant multiplied by (−1)i+j(-1)^{i+j}. Choosing a row/column with zeros simplifies the computation.

About Determinant

The determinant is a scalar value computed from a square matrix that encodes important geometric and algebraic information. For a 2×22 \times 2 matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix}, the determinant is ad−bcad - bc. A nonzero determinant means the matrix is invertible.

Learn more about Determinant →

More Determinant Examples

Example 1 easy

Find [formula].

Example 3 easy

Find [formula].

Example 4 medium

Is [formula] invertible?

← All Determinant Examples Determinant Concept