Question

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{ccc}1 & -2 & 1 \\ 5 & 5 & 4 \\ 4 & 0 & 0\end{array}\right]\)

Short answer

The determinant of the matrix is -52.

Step-by-step solution

Understanding the Matrix

We have the matrix: \[ A = \begin{bmatrix} 1 & -2 & 1 \ 5 & 5 & 4 \ 4 & 0 & 0 \end{bmatrix} \] This is a 3x3 matrix, and we need to find its determinant using cofactor expansion along the first row.

Applying the Cofactor Expansion Formula

The formula for the determinant of a 3x3 matrix \(A\) using cofactors along the first row is: \[ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \] where \(C_{ij}\) is the cofactor of element \(a_{ij}\). For our matrix, this becomes \[ \det(A) = 1 \cdot C_{11} + (-2) \cdot C_{12} + 1 \cdot C_{13} \]

Finding Minors and Cofactors

1. **Minor for** \(a_{11} = 1\): The minor is \[ \begin{vmatrix} 5 & 4 \ 0 & 0 \end{vmatrix} = (5 \cdot 0) - (4 \cdot 0) = 0 \] Thus, \(C_{11} = 0\).2. **Minor for** \(a_{12} = -2\): The minor is \[ \begin{vmatrix} 5 & 4 \ 4 & 0 \end{vmatrix} = (5 \cdot 0) - (4 \cdot 4) = -16 \] Thus, \(C_{12} = -1 \cdot (-16) = 16\).3. **Minor for** \(a_{13} = 1\): The minor is \[ \begin{vmatrix} 5 & 5 \ 4 & 0 \end{vmatrix} = (5 \cdot 0) - (5 \cdot 4) = -20 \] Thus, \(C_{13} = -20\).

Calculate the Determinant

Substitute the cofactors back into the cofactor expansion formula: \[ \det(A) = 1 \cdot 0 + (-2) \cdot 16 + 1 \cdot (-20) \]This simplifies to: \[ \det(A) = 0 - 32 - 20 = -52 \]

Conclusion

By using cofactor expansion along the first row, the determinant of the matrix is found to be \(-52\).

Key concepts

Understanding Cofactor Expansion

Cofactor expansion is a method to calculate the determinant of a square matrix. It involves breaking down the matrix into smaller matrices, making the computation more manageable. When computing the determinant of a 3x3 matrix, you can expand along any row or column, but in this instance, we'll proceed with the first row for simplicity.
To expand using the first row, you'll use the cofactor expansion formula:

• The element in the first row and first column, denoted as \(a_{11}\), multiplies its cofactor \(C_{11}\).
• You do the same for \(a_{12}\) and \(a_{13}\), incorporating their respective cofactors \(C_{12}\) and \(C_{13}\).

The cofactor itself is calculated as the minor of the element, adjusted by a prefix of \((-1)^{i+j}\), where \(i\) and \(j\) are the indices of the element. This sign adjustment is essential as it reflects the determinant's properties across different elements.

Exploring Matrix Minors

Matrix minors play a key role in finding determinants using cofactor expansion. A minor is the determinant of a smaller matrix, derived by removing one row and one column from the larger matrix.
Consider finding the minor of the element at position \(a_{11}\), which means deleting the first row and first column from the main matrix. The remaining 2x2 matrix is the one we consider for the minor. The determinant of this 2x2 matrix is the required minor.
For example, the matrix:\[\begin{bmatrix}5 & 5 \4 & 0\end{bmatrix}\]is used to compute the minor of \(a_{13}\) in the original 3x3 matrix. We calculate the determinant of this reduced matrix as \((5 \times 0) - (5 \times 4)\), giving a result of \(-20\). Remember, calculating minors requires understanding how these smaller matrices are constructed and manipulated.

Computation of Determinants

The computation of a determinant is essentially assembling all the parts coming from the cofactor expansion. Once each cofactor is calculated, you multiply each cofactor by its corresponding element from the matrix's first row, as described.
The formula looks like this: \[ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \]You substitute the known values:

• \(a_{11} = 1\), with \(C_{11} = 0\)
• \(a_{12} = -2\), with \(C_{12} = 16\)
• \(a_{13} = 1\), with \(C_{13} = -20\)

Bringing them together, we find the determinant:\[\det(A) = 1 \cdot 0 - 2 \cdot 16 + 1 \cdot (-20)\] This simplifies to:\[\det(A) = 0 - 32 - 20 = -52\]The determinant of the matrix is calculated as \(-52\), providing insights into properties like matrix invertibility and eigenvalues.