Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{cccc}-5 & -5 & 0 & -2 \\ 0 & 0 & 5 & 0 \\ 1 & 3 & 3 & 1 \\\ -4 & -2 & -1 & -5\end{array}\right]\)

Short answer

The determinant of the matrix is \(-40\).

Step-by-step solution

Selecting a Row for Expansion

To find the determinant using cofactor expansion, we can choose any row or column. Here, we choose the second row for cofactor expansion because it contains two zero elements, which simplifies calculation.

Calculating the Cofactors

For the second row, calculate the cofactor for the non-zero element at position (2,3), which is 5. The minor of this element is obtained by deleting row 2 and column 3 from the matrix. This leaves us with the smaller matrix: \[\begin{aligned}\begin{bmatrix}-5 & -5 & -2 \1 & 3 & 1 \-4 & -2 & -5\end{bmatrix}\end{aligned}\]The cofactor is then calculated as \((-1)^{2+3} \times ext{determinant of the minor} = -\text{determinant of the minor} \).

Expanding the 3x3 Minor

Calculate the determinant of the 3x3 matrix:\[\begin{aligned}-5 & -5 & -2 \1 & 3 & 1 \-4 & -2 & -5\end{aligned}\]Use cofactor expansion along the first row for simplicity:\(-5 \times \det\begin{bmatrix}3 & 1 \-2 & -5\end{bmatrix} - (-5) \times \det\begin{bmatrix}1 & 1 \ \-4 & -5\end{bmatrix} + (-2) \times \det\begin{bmatrix}1 & 3 \ \-4 & -2\end{bmatrix}\).

Calculating the Determinants of 2x2 Matrices

Evaluate each determinant from Step 3:1. \(\det\begin{bmatrix}3 & 1 \-2 & -5\end{bmatrix} = (3)(-5) - (1)(-2) = -15 + 2 = -13\)2. \(\det\begin{bmatrix}1 & 1 \ \-4 & -5\end{bmatrix} = (1)(-5) - (1)(-4) = -5 + 4 = -1\)3. \(\det\begin{bmatrix}1 & 3 \ \-4 & -2\end{bmatrix} = (1)(-2) - (3)(-4) = -2 + 12 = 10\)

Completing the Cofactor Expansion

Substitute the results from Step 4 into the expression found in Step 3:\(-5 \times (-13) - (-5) \times (-1) + (-2) \times 10\)Simplifying gives:\(-5 \times -13 = 65\),\(-5 \times -1 = 5\),\(-2 \times 10 = -20\).Add these results: \(65 - 5 + (-20) = 40\).

Calculating the Final Determinant

The cofactor was calculated for the element 5 from the original 4x4 matrix, resulting in a determinant contribution of \(-40\). Since the original determinant is expanded along that second row element, the final determinant of the given 4x4 matrix is \(-40\).

Key concepts

Cofactor Expansion

Cofactor expansion is a fundamental method for calculating the determinant of a matrix. It helps convert a larger matrix problem into manageable smaller matrix problems.
To perform cofactor expansion, we select any row or column. This choice is strategic, often picking the one with the most zeros, to simplify our calculations.
The determinant is computed by multiplying each element of the chosen row or column by its corresponding cofactor.

• The cofactor is linked to a minor, which is comprised of deleting the element's row and column and computing the determinant of the resulting smaller matrix.
• It is adjusted by a sign based on the element's position, \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column numbers.

Understanding cofactor expansion helps decompose a large scale problem into simpler subcomponents, making determinant calculation feasible.

Matrix Algebra

Matrix algebra is the study of matrices and the various rules that define operations on them. It includes fundamental tasks such as addition, subtraction, multiplication, and operations associated with finding determinants.
Matrix operations adhere to rules and laws, such as the distributive and associative laws, which are similar to those in standard arithmetic.

• Determinant calculation allows us to understand matrix behavior better, as determinants can give insights into the matrix's properties, such as invertibility.
• An important aspect learned through matrix algebra operations is recognizing patterns that enable simplification, like selecting optimal rows and columns for cofactor expansion.

Matrix algebra serves as a robust framework for solving mathematical problems involving systems of equations and other applications, which permeates into fields like engineering and physics.

Minor of a Matrix

A minor of a matrix is a crucial concept when performing cofactor expansions. It is obtained by removing a specific row and column from the matrix to form a smaller square matrix. For example, if we are considering an element in a 4x4 matrix, removing its row and column leaves us with a 3x3 matrix.
The determinant of this smaller matrix is called the minor.

• In cofactor expansions, the minor's determinant is necessary for calculating the cofactor related to each element.
• Usually, the minor simplifies the original problem, serving as a stepping stone in determining the larger matrix's determinant.

This crucial stepping is seen when transitioning from a larger matrix problem to smaller parts, making the understanding of minors essential for accuracy and efficiency.

4x4 Matrices

4x4 matrices introduce complexity that often requires methodical approaches to simplify calculations, such as determinant finding. These matrices consist of 16 elements, posing a significant computational effort for determinant calculation without a strategic approach.

• Cofactor expansion along a row or column with multiple zeroes is a preferred technique. This reduces the burden of calculating multiple smaller matrices.
• Using the structure within a 4x4 matrix, like identifying identical or zero elements, can provide strategic advantages in simplifying steps involved in finding the determinant.

These strategies allow one to transform complex matrix operations into manageable subtasks, making 4x4 matrices less daunting and more approachable, especially in introductory linear algebra courses.