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MatricesLinear algebraCollegeabout 0:50

Find the determinant of [[1,2,0,1],[0,1,3,2],[2,0,1,1],[1,1,1,1]]: animated lesson

A narrated, step-by-step matrices video lesson with captions. Final answer: det(A) = -2.

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 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    det⁡[1201013220111111]\det \begin{bmatrix}1 & 2 & 0 & 1 \\ 0 & 1 & 3 & 2 \\ 2 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1\end{bmatrix}
  2. 2. Steps, part 1
    1. A=[1201013220111111]A = \begin{bmatrix}1 & 2 & 0 & 1 \\ 0 & 1 & 3 & 2 \\ 2 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1\end{bmatrix}
      Read the 4×4 matrix
    2. Reduce to upper-triangular form with row operations
    3. [120101320−41−10−110]\begin{bmatrix}1 & 2 & 0 & 1 \\ 0 & 1 & 3 & 2 \\ 0 & -4 & 1 & -1 \\ 0 & -1 & 1 & 0\end{bmatrix}
      Eliminate below the pivot in column 1 (adding row multiples keeps the determinant)
    4. [12010132001370042]\begin{bmatrix}1 & 2 & 0 & 1 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 13 & 7 \\ 0 & 0 & 4 & 2\end{bmatrix}
      Eliminate below the pivot in column 2 (adding row multiples keeps the determinant)
  3. 3. Steps, part 2
    1. [1201013200137000−213]\begin{bmatrix}1 & 2 & 0 & 1 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 13 & 7 \\ 0 & 0 & 0 & -\frac{2}{13}\end{bmatrix}
      Eliminate below the pivot in column 3 (adding row multiples keeps the determinant)
    2. det⁡A=(1)(1)(13)(−213)=−2\det A = (1)(1)(13)(-\frac{2}{13}) = -2
      Multiply the diagonal of the triangular matrix
  4. 4. The answer
    det⁡A=−2\det A = -2

    det = -2

  5. 5. Check
    • Computed exactly with fractions; row reduction gives the same value -2.

Transcript

The problem. Here is the problem: [[1, 2, 0, 1], [0, 1, 3, 2], [2, 0, 1, 1], [1, 1, 1, 1]]. Find the determinant of a 4 times 4 matrix.

Steps, part 1. Read the 4 times 4 matrix. Reduce to upper-triangular form with row operations. Eliminate below the pivot in column 1 (adding row multiples keeps the determinant). Eliminate below the pivot in column 2 (adding row multiples keeps the determinant).

Steps, part 2. Eliminate below the pivot in column 3 (adding row multiples keeps the determinant). Multiply the diagonal of the triangular matrix.

The answer. So the answer is det(A) equals negative 2. The determinant is negative 2, which is non-zero, so the matrix is invertible and its rows are linearly independent.

Check. Computed exactly with fractions; row reduction gives the same value negative 2.

About matrices

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

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