Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}-4 & 4 & -4 \\ 0 & 0 & -3 \\ -2 & -2 & -1\end{array}\right]\)

Short answer

The determinant of the matrix is -48.

Step-by-step solution

Choose a Row or Column

To apply the cofactor expansion, we'll choose a row or column. In this case, let's use the first row \([-4, 4, -4]\), since it has small numbers which may simplify calculations.

Calculate the Cofactor of the First Element

Find the cofactor of the first element \(-4\). To do this, we remove the first row and first column, leaving the submatrix \left[\begin{array}{cc}0 & -3 \ -2 & -1\end{array}\right]\, whose determinant is computed as \(0 \times (-1) - (-2) \times (-3) = 6\). The cofactor is \-1^{1+1}(-4)(6) = -24\.

Calculate the Cofactor of the Second Element

For the second element, \(4\), remove the first row and second column, leaving \(\left[\begin{array}{cc}0 & -3 \ -2 & -1\end{array}\right]\). Calculating its determinant gives \(0 \times (-1) - (-2) \times (-3) = 6\). Therefore, the cofactor is \-1^{1+2}(4)(6) = -24\.

Calculate the Cofactor of the Third Element

For the third element, \(-4\), remove the first row and third column to get \(\left[\begin{array}{cc}0 & 0 \ -2 & -2\end{array}\right]\). Calculate its determinant: \(0 \times (-2) - 0 \times (-2) = 0\), hence the cofactor is \-1^{1+3}(-4)(0) = 0\.

Sum the Cofactors to Find the Determinant

Now, sum all the cofactor contributions: \(-24 - 24 + 0 = -48\). Therefore, the determinant of the matrix is \(-48\).

Key concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a technique used to calculate the determinant of a matrix. The process relies on breaking down a larger matrix into smaller, more manageable pieces. By focusing on a particular row or column, we can solve for each element's cofactor. Each cofactor is the determinant of a smaller matrix formed by removing the row and column of the element in question. Then, we alternate the signs of these cofactors based on their position within the original matrix.
To keep things simple, remember:

• Choose a row or column. Often, one with zeros or smaller numbers is easiest.
• The sign is determined by \((-1)^{i+j}\), where i and j are the element's row and column numbers, respectively.
• Summing up all these cofactor contributions gives you the determinant.

This approach is particularly useful for 3x3 matrices because it reduces complexity, helping you focus on smaller parts of the matrix at a time.

3x3 Matrix

When calculating determinants, a 3x3 matrix is a square matrix with three rows and three columns. It looks something like this:\[ \begin{bmatrix}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33}\end{bmatrix}\]Understanding how to handle a 3x3 matrix is key in linear algebra because it serves as the foundation for larger matrices as well. Determinants help in various operations, like solving equations, finding out if a matrix has an inverse, and characterizing its properties.
A good practice when dealing with 3x3 matrices is to check if any rows or columns contain zeros, which simplifies calculations drastically. This is because zero-cofactors mean less arithmetic work, something highly advantageous when performing cofactor expansion.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. Matrices and determinants are fundamental parts of linear algebra. They help to describe linear equations, transformations, and systems.
Determinants in linear algebra are used to:

• Determine if a matrix is invertible (only non-zero determinants).
• Describe the volume scaling factor of linear transformations represented by matrices.
• Analyze systems of linear equations for consistency and uniqueness of solutions.

Thus, grasping concepts like cofactor expansion for easy determinant calculation is crucial. It aids in understanding the broader capabilities provided by matrices in linear algebra, impacting fields such as engineering, physics, and even computer science.

Matrix Algebra

Matrix algebra involves the study and application of operations on matrices. It is a vital area of mathematics as it lays the groundwork for more complex algebraic calculations and mathematical modeling. Determinant calculation is one part of matrix algebra.
In matrix algebra, the determinant of a matrix can also give insights into:

• Singularity: A matrix with a zero determinant is singular and does not have an inverse.
• Transformations: Determinants can describe scaling and rotations represented by matric transformations.
• Eigenvalues: They are closely linked to the characteristic equation of a matrix.

Matrix algebra extends beyond theoretical mathematics. It greatly influences computational fields, like computer graphics, cryptography, and machine learning. Understanding how to calculate determinants and interpret their meaning is vital across disciplines.