Math · Precalculus · Worked example

Find the determinant of a 3 × 3 matrix

Find the determinant of the matrix A with rows (2, 1, 3), (0, −1, 4) and (1, 2, 0). Does A have an inverse?

A = (\begin{array}{cccccccccc} 2 & 1 & 3 \\ 0 & - 1 & 4 \\ 1 & 2 & 0 \end{array})

Expand along the first row

Multiply each entry of row 1 by the determinant left when you delete its row and column, with the signs +, −, +.

det A = 2 | \begin{array}{cccccccccc} - 1 & 4 \\ 2 & 0 \end{array} | - 1 | \begin{array}{cccccccccc} 0 & 4 \\ 1 & 0 \end{array} | + 3 | \begin{array}{cccccccccc} 0 & - 1 \\ 1 & 2 \end{array} |

\begin{aligned} det A & = 2 | \begin{array}{cccccccccc} - 1 & 4 \\ 2 & 0 \end{array} | - 1 | \begin{array}{cccccccccc} 0 & 4 \\ 1 & 0 \end{array} | \\ + 3 | \begin{array}{cccccccccc} 0 & - 1 \\ 1 & 2 \end{array} | \end{aligned}

Evaluate the 2 × 2 determinants

Each one is ad − bc.

\begin{array}{l} (- 1) (0) - (4) (2) = - 8 \\ (0) (0) - (4) (1) = - 4 \\ (0) (2) - (- 1) (1) = 1 \end{array}

Combine

2 (- 8) - 1 (- 4) + 3 (1) = - 9

Interpret

The determinant is −9, not 0, so A has an inverse. As a transformation of space, A scales volume by 9 and reverses orientation.

Result

det A = −9, so A has an inverse.

Your turn

Find the determinant of the matrix with rows (4, −2) and (3, 5).

Show the answer and explanation

26.

ad − bc = 4 · 5 − (−2)(3) = 20 + 6 = 26.

4 \cdot 5 - (- 2) \cdot 3 = 26

Keep exploring

In Matrices & linear systems, the determinant reads −9. Change the last row to (2, 0, 7), the sum of the first two rows, and the determinant becomes 0: that matrix has no inverse.

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