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MatricesLinear algebraCollegeabout 0:43

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

A narrated, step-by-step matrices video lesson with captions. Final answer: det(A) = 238.

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 (0.3 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    det⁡[−32−5−7−1−6−923]\det \begin{bmatrix}-3 & 2 & -5 \\ -7 & -1 & -6 \\ -9 & 2 & 3\end{bmatrix}
  2. 2. Step by step
    1. A=[−32−5−7−1−6−923]A = \begin{bmatrix}-3 & 2 & -5 \\ -7 & -1 & -6 \\ -9 & 2 & 3\end{bmatrix}
      Read the 3×3 matrix
    2. det⁡A=(−3)∣−1−623∣−(2)∣−7−6−93∣+(−5)∣−7−1−92∣\det A = (-3)\begin{vmatrix}-1 & -6 \\ 2 & 3\end{vmatrix} - (2)\begin{vmatrix}-7 & -6 \\ -9 & 3\end{vmatrix} + (-5)\begin{vmatrix}-7 & -1 \\ -9 & 2\end{vmatrix}
      Expand along the first row (cofactor expansion)
    3. M11=9,M12=−75,M13=−23M_{11} = 9,\quad M_{12} = -75,\quad M_{13} = -23
      Evaluate each 2×2 minor
    4. det⁡A=(−3)(9)−(2)(−75)+(−5)(−23)=238\det A = (-3)(9) - (2)(-75) + (-5)(-23) = 238
      Combine with the alternating signs + − +
  3. 3. The answer
    det⁡A=238\det A = 238

    det = 238

  4. 4. Check
    • Computed exactly with fractions; row reduction gives the same value 238.

Transcript

The problem. Here is the problem: [[-3, 2, negative 5], [-7, negative 1, negative 6], [-9, 2, 3]]. 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 238. The determinant is 238, 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 238.

About matrices

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

More lessons like this

Find the determinant of [[-3,2,-5],[-7,-1,-6],[-9,2,3]]