Questions: Find the determinant of this (3 times 3) matrix using expansion by minors about the first column. [ A=left[ beginarrayrrr 9 -2 9 5 4 9 9 1 7 endarray right] ] [ A=left beginarrayrrr 9 -2 9 5 4 9 9 1 7 endarray right ]

Solution Steps

To find the determinant of a \(3 \times 3\) matrix using expansion by minors about the first column, follow these steps:

1. Select the first column of the matrix for expansion.
2. For each element in the first column, calculate the minor by removing the row and column of that element.
3. Compute the determinant of each \(2 \times 2\) minor.
4. Apply the cofactor expansion formula: \( |A| = a_{11}C_{11} + a_{21}C_{21} + a_{31}C_{31} \), where \( C_{ij} \) is the cofactor of element \( a_{ij} \).

We are given the matrix \( A \) as follows: \[ A = \begin{bmatrix} 9 & -2 & 9 \\ 5 & 4 & 9 \\ 9 & 1 & 7 \end{bmatrix} \]

To find the determinant \( |A| \) using expansion by minors about the first column, we calculate: \[ |A| = a_{11}C_{11} + a_{21}C_{21} + a_{31}C_{31} \] where \( a_{ij} \) are the elements of the first column and \( C_{ij} \) are the corresponding cofactors.

1. For \( a_{11} = 9 \): \[ C_{11} = \begin{vmatrix} 4 & 9 \\ 1 & 7 \end{vmatrix} = (4)(7) - (9)(1) = 28 - 9 = 19 \] Thus, \( a_{11}C_{11} = 9 \cdot 19 = 171 \).

2. For \( a_{21} = 5 \): \[ C_{21} = -\begin{vmatrix} -2 & 9 \\ 1 & 7 \end{vmatrix} = -((-2)(7) - (9)(1)) = -(-14 - 9) = 23 \] Thus, \( a_{21}C_{21} = 5 \cdot 23 = 115 \).

3. For \( a_{31} = 9 \): \[ C_{31} = \begin{vmatrix} -2 & 9 \\ 4 & 9 \end{vmatrix} = (-2)(9) - (9)(4) = -18 - 36 = -54 \] Thus, \( a_{31}C_{31} = 9 \cdot (-54) = -486 \).

Now, we combine the results to find the determinant: \[ |A| = 171 - 115 - 486 = -430 \]

Final Answer