Cofactor Expansion Exercise

Question #1

\[\] \[ \text{Given the matrix } A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Question #2

\[ \text{Given the matrix } B = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 0 & 5 \\ 7 & 6 & 8 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Question #3

  \[ \text{Given the matrix } C = \begin{pmatrix} 3 & 2 & 4 \\ 1 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Question #4

\[ \text{Given the matrix } D = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Solution of Exercise

Question #1

\[ \text{Given the matrix } A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Solution:

\[ \text{We will expand along the first row. The determinant is:} \] \[ \text{det}(A) = 1 \cdot \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} – 2 \cdot \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} + 3 \cdot \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} \] \[ \text{Now, calculate the 2×2 determinants:} \] \[ \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} = (5 \times 9) – (6 \times 8) = 45 – 48 = -3 \] \[ \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = (4 \times 9) – (6 \times 7) = 36 – 42 = -6 \] \[ \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = (4 \times 8) – (5 \times 7) = 32 – 35 = -3 \] \[ \text{Substitute these values back into the cofactor expansion:} \] \[ \text{det}(A) = 1 \cdot (-3) – 2 \cdot (-6) + 3 \cdot (-3) \] \[ \text{det}(A) = -3 + 12 – 9 \] \[ \text{det}(A) = 0 \] \[ \textbf{Answer: The determinant of matrix } A \text{ is } 0. \]

Question #2

\[ \text{Given the matrix } B = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 0 & 5 \\ 7 & 6 & 8 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Solution:

\[ \text{We will expand along the first row. The determinant is:} \] \[ \text{det}(B) = 2 \cdot \begin{vmatrix} 0 & 5 \\ 6 & 8 \end{vmatrix} – 3 \cdot \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} + 1 \cdot \begin{vmatrix} 4 & 0 \\ 7 & 6 \end{vmatrix} \] \[ \text{Now, calculate the 2×2 determinants:} \] \[ \begin{vmatrix} 0 & 5 \\ 6 & 8 \end{vmatrix} = (0 \times 8) – (5 \times 6) = 0 – 30 = -30 \] \[ \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = (4 \times 8) – (5 \times 7) = 32 – 35 = -3 \] \[ \begin{vmatrix} 4 & 0 \\ 7 & 6 \end{vmatrix} = (4 \times 6) – (0 \times 7) = 24 – 0 = 24 \] \[ \text{Substitute these values back into the cofactor expansion:} \] \[ \text{det}(B) = 2 \cdot (-30) – 3 \cdot (-3) + 1 \cdot 24 \] \[ \text{det}(B) = -60 + 9 + 24 \] \[ \text{det}(B) = -27 \] \[ \textbf{Answer: The determinant of matrix } B \text{ is } -27. \]      

Question #3

  \[ \text{Given the matrix } C = \begin{pmatrix} 3 & 2 & 4 \\ 1 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Solution:

\[ \text{We will expand along the first row. The determinant is:} \] \[ \text{det}(C) = 3 \cdot \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} – 2 \cdot \begin{vmatrix} 1 & 6 \\ 7 & 9 \end{vmatrix} + 4 \cdot \begin{vmatrix} 1 & 5 \\ 7 & 8 \end{vmatrix} \] \[ \text{Now, calculate the 2×2 determinants:} \] \[ \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} = (5 \times 9) – (6 \times 8) = 45 – 48 = -3 \] \[ \begin{vmatrix} 1 & 6 \\ 7 & 9 \end{vmatrix} = (1 \times 9) – (6 \times 7) = 9 – 42 = -33 \] \[ \begin{vmatrix} 1 & 5 \\ 7 & 8 \end{vmatrix} = (1 \times 8) – (5 \times 7) = 8 – 35 = -27 \] \[ \text{Substitute these values back into the cofactor expansion:} \] \[ \text{det}(C) = 3 \cdot (-3) – 2 \cdot (-33) + 4 \cdot (-27) \] \[ \text{det}(C) = -9 + 66 – 108 \] \[ \text{det}(C) = -51 \] \[ \textbf{Answer: The determinant of matrix } C \text{ is } -51. \]      

Question #4

\[ \text{Given the matrix } D = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix}, \text{ compute the determinant using cofactor expansion.} \]

Solution:

\[ \text{We will expand along the first row. The determinant is:} \] \[ \text{det}(D) = 1 \cdot \begin{vmatrix} 5 & 8 \\ 6 & 9 \end{vmatrix} – 4 \cdot \begin{vmatrix} 2 & 8 \\ 3 & 9 \end{vmatrix} + 7 \cdot \begin{vmatrix} 2 & 5 \\ 3 & 6 \end{vmatrix} \] \[ \text{Now, calculate the 2×2 determinants:} \] \[ \begin{vmatrix} 5 & 8 \\ 6 & 9 \end{vmatrix} = (5 \times 9) – (8 \times 6) = 45 – 48 = -3 \] \[ \begin{vmatrix} 2 & 8 \\ 3 & 9 \end{vmatrix} = (2 \times 9) – (8 \times 3) = 18 – 24 = -6 \] \[ \begin{vmatrix} 2 & 5 \\ 3 & 6 \end{vmatrix} = (2 \times 6) – (5 \times 3) = 12 – 15 = -3 \] \[ \text{Substitute these values back into the cofactor expansion:} \] \[ \text{det}(D) = 1 \cdot (-3) – 4 \cdot (-6) + 7 \cdot (-3) \] \[ \text{det}(D) = -3 + 24 – 21 \] \[ \text{det}(D) = 0 \] \[ \textbf{Answer: The determinant of matrix } D \text{ is } 0. \]

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