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Find the determinant of [[1,2,3],[4,5,6],[7,8,9]]: animated lesson

A narrated, step-by-step matrices video lesson with captions. Final answer: det(A) = 0.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.2 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\det \begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$
2. Step by step
1. $A = \begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$
  Read the 3×3 matrix
2. $\det A = (1)\begin{vmatrix}5 & 6 \\ 8 & 9\end{vmatrix} - (2)\begin{vmatrix}4 & 6 \\ 7 & 9\end{vmatrix} + (3)\begin{vmatrix}4 & 5 \\ 7 & 8\end{vmatrix}$
  Expand along the first row (cofactor expansion)
3. $M_{11} = -3,\quad M_{12} = -6,\quad M_{13} = -3$
  Evaluate each 2×2 minor
4. $\det A = (1)(-3) - (2)(-6) + (3)(-3) = 0$
  Combine with the alternating signs + − +
3. The answer
$\det A = 0$
det = 0
4. Check
- Computed exactly with fractions; row reduction gives the same value 0.

Transcript

The problem. Here is the problem: [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. Find the determinant of a 3 times 3 matrix.

Step by step. Read the 3 times 3 matrix. Expand along the first row (cofactor expansion). Evaluate each 2 times 2 minor. Combine with the alternating signs plus minus +.

The answer. So the answer is det(A) equals 0. The determinant is 0, so the matrix is singular: it has no inverse and its rows are linearly dependent.

Check. Computed exactly with fractions; row reduction gives the same value 0.

About matrices

Determinants, inverses, products, transposes and eigenvalues of matrices.

Find the determinant of [[1,2,3],[4,5,6],[7,8,9]] (Lesson)