Question

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

Short answer

The determinant of the matrix is 0.

Step-by-step solution

Determine Matrix Size and Choose Expansion

The given matrix is a 3x3 matrix. We can use cofactor expansion along any row or column. To simplify the calculations, we'll expand along the second row, which contains all zeros. This row will make the computation straightforward because all of its cofactors will be multiplied by zero.

Cofactor Expansion

For the matrix \(\begin{bmatrix}-4 & 1 & 1 \ 0 & 0 & 0 \ -1 & -2 & -5\end{bmatrix}\), expand the determinant using the second row: \[ \text{det}(A) = 0 \cdot C_{21} + 0 \cdot C_{22} + 0 \cdot C_{23} = 0 \] Since the entire row is zero, each term in the expansion is zero.

Conclusion of Determinant Calculation

Since all terms are multiplied by zero, the determinant of the matrix is zero. This is expected because the second row being entirely zero implies that the rows of the matrix are linearly dependent.

Key concepts

Understanding Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate determinants of matrices. Imagine it as breaking down a larger problem into smaller, more manageable ones. With cofactor expansion, you can use any row or column of the matrix to compute its determinant.
For each element in the chosen row or column, you create a smaller matrix by removing the current element's row and column. The determinant of this smaller matrix is then multiplied by the original matrix element and a sign factor. These results are summed to get the determinant of the original matrix.
This method becomes extremely useful for larger matrices, as it allows you to select rows or columns that simplify calculations, like a row or column full of zeros.

3x3 Matrices: A Basic Overview

A 3x3 matrix contains three rows and three columns, making up nine elements. It can be represented as:

Such matrices are commonly used in various mathematical and engineering problems. They are small enough for hand calculations, which is why they're a great stepping stone to understanding more complex matrices.
In a 3x3 matrix, calculate the determinant with cofactor expansion or the cross-product method, especially in physics problems. Each entry plays a critical role in determining the matrix's properties.
In our example, choosing to expand along the second row, which contains all zeros, instantly simplifies the determinant calculation to zero.

Linear Dependence in Matrices

Linear dependence occurs when one row or column in a matrix can be expressed as a combination of others. This concept is fundamental in understanding matrix behavior.
When a matrix has a row or column entirely made of zeros, it indicates linear dependence. This is because you can form this row by multiplying the other rows by zero. If any row is a linear combination of others, the determinant of the matrix is zero.
In the given matrix, the presence of the zero row implies linear dependence, proving why the determinant equals zero. Linear dependence is a key aspect when analyzing systems of linear equations and influences whether solutions are unique.

Exploring Matrix Algebra

Matrix algebra is an area of mathematics focusing on the manipulation of matrices. It involves operations such as addition, subtraction, multiplication, and finding determinants. Matrix algebra simplifies the handling of large datasets and solving complex equations.
Understanding matrix operations is crucial for fields like physics, engineering, and computer science. For example, matrix multiplication is used in transformations, while finding the determinant can indicate the matrix's invertibility.
Our example emphasizes the determinant's importance. A zero determinant suggests the matrix doesn't have an inverse, highlighting the limitations and dependencies present in matrix systems.