Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}-2 & -3 & 5 \\ 5 & 2 & 0 \\ -1 & 0 & 0\end{array}\right]\)

Short answer

The determinant of the matrix is 10.

Step-by-step solution

Choose the Row or Column for Expansion

We will perform cofactor expansion along the third column of the matrix. This column contains two zeros, which will simplify the computation. The third column is \( \begin{bmatrix} 5 \ 0 \ 0 \end{bmatrix} \).

Set Up the Expansion Formula

The determinant of a 3x3 matrix using cofactor expansion along the third column can be written as:\[\text{det}(A) = c_{13}A_{13} + c_{23}A_{23} + c_{33}A_{33}\]where \( c_{ij} \) is the element in the \(i^{th}\) row and \(j^{th}\) column, and \( A_{ij} \) is the cofactor of \( c_{ij} \).

Calculate the Cofactors

To calculate the cofactor, remove the row and column of each element.- For \( c_{13} = 5 \): - Cofactor \( A_{13} = (-1)^{1+3} \times \begin{vmatrix} 5 & 2 \ -1 & 0 \end{vmatrix} = 1 \times (5 \times 0 - 2 \times (-1)) = 2 \).- For \( c_{23} = 0 \), the cofactor is irrelevant since it will multiply by zero.- For \( c_{33} = 0 \), the cofactor is also irrelevant since it will multiply by zero.

Evaluate the Determinant

Using the cofactor expansion formula:\[\text{det}(A) = 5 \times 2 + 0 \times A_{23} + 0 \times A_{33} = 10\]Thus, the determinant of the matrix is 10.

Key concepts

Cofactor Expansion

Cofactor expansion is a commonly used technique for computing the determinant of a matrix. It involves expanding the determinant along a row or column of your choosing.
In this method, selecting a row or column that contains zeroes can significantly simplify calculations. This is because any term multiplied by zero will have no impact on the final result.
In our example, the third column was chosen for cofactor expansion because it has the elements \[\begin{bmatrix} 5 \ 0 \ 0 \end{bmatrix}\], making the evaluation relatively straightforward.

• Only one non-zero element, 5, needs a cofactor computation.
• Zero elements make two terms vanish in the expansion formula.

As a result, our workload significantly decreases, only requiring calculation for the single active cofactor.

Cofactor Calculation

Calculating the cofactor is a crucial step in determining the determinant of a matrix. To find a cofactor, you need to remove the specific row and column of the element under consideration.
This leaves a smaller matrix, often a 2x2 if you start with a 3x3 matrix. Once this smaller matrix is identified, compute its determinant.
In the present example, for the element 5 located in the first row and third column, the cofactor process is:

• Remove the first row and third column.
• The smaller matrix is \[\begin{vmatrix} 5 & 2 \ -1 & 0 \end{vmatrix}\].
• The cofactor for element 5 becomes \[(-1)^{1+3} \text{det} \begin{vmatrix} 5 & 2 \ -1 & 0 \end{vmatrix} = 1 \times (5 \times 0 - 2 \times (-1)) = 2\].

This computation shows that for the element at this position, the cofactor is 2, which directly contributes to the final determinant due to it being on the chosen expansion column.

Matrix Algebra

Matrix algebra encompasses various operations that you might encounter while working with matrices, determinant calculation being one of them. It is essential because determinants can be used to solve systems of linear equations, among other applications.
The determinant is a special number that can be calculated from a matrix, often denoted as det(A) for a matrix A. When a determinant is non-zero, the matrix is invertible, meaning it has a unique inverse.
In our practical example, knowing how to find the determinant using methods like cofactor expansion helps understand more about the matrix properties and facilitates solving related algebra problems. Matrix algebra forms a backbone for many advanced topics, providing foundational tools for deeper exploration into areas like linear transformations and eigenvalues and eigenvectors.