Determinants - Solved Examples – 3×3 matrix

  • In this lecture, we will solve some examples involving determinants of 3×3 matrices.

Example 1

  • Find the value of the determinant of the matrix: [ \begin{pmatrix} 2 & 1 & 4 \ 3 & 0 & -2 \ -1 & 3 & 1 \ \end{pmatrix} ]

  • To compute the determinant, we can use the formula: [ \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} ] Where (a_{ij}) represents the entry in the (i)th row and (j)th column, and (C_{ij}) represents the cofactor of (a_{ij}).

  • Let’s calculate the cofactors:

    • (C_{11} = (-1)^{1+1} \cdot \text{det}(M_{11}))
      • (M_{11} = \begin{pmatrix} 0 & -2 \ 3 & 1 \end{pmatrix})
      • (\text{det}(M_{11}) = 0 \cdot 1 - (-2) \cdot 3 = 6)
      • (C_{11} = (-1)^{1+1} \cdot 6 = 6)
    • Similarly, we can find (C_{12}) and (C_{13}).

  • Using the determinants of the cofactor matrices, we can now calculate the determinant of the original matrix: [ \text{det}(A) = 2(6) + 1C_{12} + 4C_{13} ]

  • Thus, the value of the determinant is (12 + C_{12} + 4C_{13}).

Example 2

  • Find the value of the determinant of the matrix: [ \begin{pmatrix} -3 & 0 & 2 \ 1 & -2 & 0 \ 4 & 1 & -1 \ \end{pmatrix} ]

  • Using the formula for the determinant, we have: [ \text{det}(A) = -3C_{11} + 0C_{12} + 2C_{13} ]

  • To calculate (C_{11}):

    • (M_{11} = \begin{pmatrix} -2 & 0 \ 1 & -1 \end{pmatrix})
    • (\text{det}(M_{11}) = -2 \cdot (-1) - 0 \cdot 1 = 2)
    • (C_{11} = (-1)^{1+1} \cdot \text{det}(M_{11}) = 2)

  • Using the determinants of the cofactor matrices, we can now calculate the determinant of the original matrix: [ \text{det}(A) = -3(2) + 0C_{12} + 2C_{13} ]

  • Thus, the value of the determinant is (-6 + 2C_{13}).

Example 3

  • Find the value of the determinant of the matrix: [ \begin{pmatrix} 1 & 2 & 1 \ 1 & -1 & 0 \ 3 & 1 & -2 \ \end{pmatrix} ]

  • Using the formula for the determinant, we have: [ \text{det}(A) = 1C_{11} - 2C_{12} + 1C_{13} ]

  • To calculate (C_{11}):

    • (M_{11} = \begin{pmatrix} -1 & 0 \ 1 & -2 \end{pmatrix})
    • (\text{det}(M_{11}) = -1 \cdot (-2) - 0 \cdot 1 = 2)
    • (C_{11} = (-1)^{1+1} \cdot \text{det}(M_{11}) = 2)

  • To calculate (C_{12}):

    • (M_{12} = \begin{pmatrix} 1 & 0 \ 3 & -2 \end{pmatrix})
    • (\text{det}(M_{12}) = 1 \cdot (-2) - 0 \cdot 3 = -2)
    • (C_{12} = (-1)^{1+2} \cdot \text{det}(M_{12}) = -2)

  • To calculate (C_{13}):

    • (M_{13} = \begin{pmatrix} 1 & -1 \ 3 & 1 \end{pmatrix})
    • (\text{det}(M_{13}) = 1 \cdot 1 - (-1) \cdot 3 = 4)
    • (C_{13} = (-1)^{1+3} \cdot \text{det}(M_{13}) = -4)

  • Using the determinants of the cofactor matrices, we can now calculate the determinant of the original matrix: [ \text{det}(A) = 1(2) - 2(-2) + 1(-4) ]

  • Thus, the value of the determinant is (2 + 4 + (-4) = 2).
    This concludes the first three examples. We will continue solving more examples in the next set of slides.

Determinants - Solved Examples – 3×3 matrix

Slide 11:

  • Example 4
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 3 & 1 & -2 \ 0 & 2 & 1 \ -1 & -3 & 2 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 3C_{11} - 1C_{12} + (-2)C_{13} ]

Slide 12:

  • Example 5
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 5 & -1 & 3 \ 2 & 0 & -2 \ -3 & 2 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 5C_{11} - (-1)C_{12} + 3C_{13} ]

Slide 13:

  • Example 6
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 4 & 1 & 0 \ 2 & 3 & -2 \ -1 & 0 & 2 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 4C_{11} - 1C_{12} + 0C_{13} ]

Slide 14:

  • Example 7
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 2 & -1 & 3 \ 0 & 1 & -2 \ 1 & 0 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 2C_{11} - (-1)C_{12} + 3C_{13} ]

Slide 15:

  • Example 8
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} -1 & 0 & 1 \ 3 & 2 & 0 \ 0 & -2 & 1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = (-1)C_{11} - 0C_{12} + 1C_{13} ]

Slide 16:

  • Example 9
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 3 & 2 & 1 \ 0 & 1 & -2 \ -1 & 0 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 3C_{11} - 2C_{12} + 1C_{13} ]

Slide 17:

  • Example 10
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 4 & -3 & 0 \ 1 & 0 & 2 \ -2 & 1 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 4C_{11} - (-3)C_{12} + 0C_{13} ]

Slide 18:

  • Example 11
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 2 & 0 & 1 \ 1 & -3 & 0 \ 0 & -1 & 2 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 2C_{11} - 0C_{12} + 1C_{13} ]

Slide 19:

  • Example 12
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 3 & -2 & 1 \ 0 & 1 & 0 \ -1 & 0 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 3C_{11} - (-2)C_{12} + 1C_{13} ]

Slide 20:

  • Example 13
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 1 & -2 & 0 \ 0 & 3 & 1 \ -1 & 0 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 1C_{11} - (-2)C_{12} + 0C_{13} ]

Determinants - Solved Examples – 3×3 matrix

Slide 21:

  • Example 14
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 2 & 1 & 2 \ -1 & 3 & 0 \ 3 & -2 & 1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 2C_{11} - 1C_{12} + 2C_{13} ]

Slide 22:

  • Example 15
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 1 & 0 & 2 \ -2 & 1 & 1 \ 3 & -1 & 0 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 1C_{11} - 0C_{12} + 2C_{13} ]

Slide 23:

  • Example 16
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} -1 & 0 & 1 \ 0 & -2 & 0 \ 3 & -1 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = -1C_{11} - 0C_{12} + 1C_{13} ]

Slide 24:

  • Example 17
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 0 & 1 & 3 \ 2 & 3 & 0 \ -1 & 2 & -2 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 0C_{11} - 1C_{12} + 3C_{13} ]

Slide 25:

  • Example 18
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 1 & 2 & 1 \ -1 & 0 & -2 \ 0 & -3 & 2 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 1C_{11} - 2C_{12} + 1C_{13} ]

Slide 26:

  • Example 19
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 2 & -1 & 0 \ -1 & 3 & -2 \ 0 & -2 & 1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 2C_{11} - (-1)C_{12} + 0C_{13} ]

Slide 27:

  • Example 20
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 1 & -2 & 3 \ 0 & 1 & 2 \ 3 & 0 & -1 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 1C_{11} - (-2)C_{12} + 3C_{13} ]

Slide 28:

  • Example 21
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 1 & 0 & -2 \ -1 & -2 & 1 \ 3 & 1 & 0 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 1C_{11} - 0C_{12} + (-2)C_{13} ]

Slide 29:

  • Example 22
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 0 & 1 & 2 \ 3 & 0 & -1 \ -2 & 1 & 0 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 0C_{11} - 1C_{12} + 2C_{13} ]

Slide 30:

  • Example 23
    • Find the value of the determinant of the matrix: [ \begin{pmatrix} 4 & -2 & 1 \ 2 & 0 & 3 \ -1 & 1 & -2 \ \end{pmatrix} ]
    • Using the formula for the determinant: [ \text{det}(A) = 4C_{11} - (-2)C_{12} + 1C_{13} ]