Question

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

Short answer

The determinant is 373.

Step-by-step solution

Select the First Row for Expansion

Consider the first row of the matrix \( [2, -1, 4, 4] \). We will expand along this row to calculate the determinant.

Calculate Cofactor for First Element

The first element of the first row is \(-2\). The cofactor is calculated by deleting the first row and the first column, leading to the submatrix: \[ \begin{bmatrix} -3 & 3 & 2 \ 4 & -5 & 1 \ -5 & -2 & -5 \end{bmatrix} \] The determinant of this 3x3 matrix is found using the standard determinant formula for 3x3 matrices.

Calculate Determinant of 3x3 Submatrix (A)

Calculate the determinant of the submatrix: \[ \text{det}(A) = (-3)((-5)(-5) - (-2)(1)) - (3)((4)(-5) - (-5)(1)) + (2)((4)(-2) - (-5)(4)) \] Performing the calculations gives: \[ (-3)(25 + 2) - 3(-20 + 5) + 2(-8 + 20) = (-3)(27) - 3(-15) + 2(12) \]\[ = -81 + 45 + 24 = -12 \]

Calculate Cofactor for Second Element

The second element of the first row is \(-1\). Delete the first row and second column to extract the submatrix: \[ \begin{bmatrix} 3 & 3 & 2 \ 0 & -5 & 1 \ -2 & -2 & -5 \end{bmatrix} \] Calculate its determinant as follows.

Calculate Determinant of 3x3 Submatrix (B)

Calculate the determinant of the submatrix:\[ \text{det}(B) = 3((-5)(-5) - (1)(-2)) - 3((0)(-5) - (-2)(1)) + 2((0)(-2) - (-5)(-2)) \] Solving it gives: \[ 3(25 + 2) - 3(0 + 2) + 2(0 - 10) = 3(27) - 3(2) + 2(-10) \]\[ = 81 - 6 - 20 = 55 \]

Calculate Cofactor for Third Element

The third element of the first row is \(4\). The cofactor involves removing the first row and the third column, resulting in: \[ \begin{bmatrix} 3 & -3 & 2 \ 0 & 4 & 1 \ -2 & -5 & -5 \end{bmatrix} \] The determinant calculation follows.

Calculate Determinant of 3x3 Submatrix (C)

Find the determinant of the submatrix:\[ \text{det}(C) = 3(4 \cdot (-5) - 1 \cdot (-5)) + 3(0 \cdot (-5) - 1 \cdot (-2)) + 2(0 \cdot (-5) - 4 \cdot (-2)) \] Calculate: \[ 3(-20 + 5) - 2(0 + 8) + (0 - 10) \]\[ = 3(-15) + 40 \]\[ = -45 + 40 = 35 \]

Calculate Cofactor for Fourth Element

The fourth element of the first row is \(4\). Removal of the first row and the fourth column gives:\[ \begin{bmatrix} 3 & -3 & 3 \ 0 & 4 & -5 \ -2 & -5 & -2 \end{bmatrix} \] Find its determinant next.

Calculate Determinant of 3x3 Submatrix (D)

Determine the submatrix determinant: \[ 3((4)(-2) - (-5)(-5)) - (-3)((0)(-2) - (-5)(-2)) + 3((0)(-5) - 4(-2)) \] After solving: \[ (-24 - 25) - 3(10) + 24 \]\[ = -132 + 30 + 24 = 78 \]

Compute Total Determinant Using Cofactor Expansion

Utilizing the cofactors from each element of the first row, the determinant is given by: \[ 2(-12) - 1(55) + 4(35) + 4(78) \]\[ = -24 - 55 + 140 + 312 \]\[ = 373 \]

Key concepts

Matrix Algebra

Matrix algebra is a vital concept in mathematics, especially useful in solving real-world problems involving linear equations. A matrix is simply an organized collection of numbers arranged in rows and columns. Matrices can represent complex data and systems such as transformations and connections. Each number in a matrix is called an element.

In an algebraic context, operations such as addition, subtraction, and multiplication are performed on matrices. Matrix multiplication follows special rules that require the number of columns in the first matrix to match the number of rows in the second matrix. This rule ensures the operation results in a compatible product matrix.

Determinants in matrix algebra offer insights such as understanding systems of linear equations and determining their solvability. They provide crucial information about the properties of matrices, such as invertibility. If you have a matrix, you might want to compute its determinant to gain an understanding of these characteristics.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of larger matrices. It's a step-by-step approach that allows you to break down the computation into smaller, more manageable parts. This method is especially useful for matrices greater than 2x2 in size, like the 4x4 matrix in the exercise.

To perform cofactor expansion, you choose a specific row or column to expand along. This exercise chose the first row. For each element in the chosen row or column, you:

• Calculate its minor, which involves removing the row and column of that element, resulting in a submatrix.
• Compute the determinant of the resulting submatrix.
• Consider the sign based on the element's position (using "+" and "-" alternately starting with "+" for the first element).

Multiply the determinant of the submatrix by the corresponding element of the matrix and its sign, summing these products gives the overall determinant of the matrix.

Cofactor expansion is a powerful technique as it reduces the complexity of calculating determinants of large matrices into simpler calculations involving smaller submatrices. This method provides a structured approach to systematically breaking down these calculations.

Submatrix

A submatrix is created when you delete specific rows and columns from a larger matrix. In the context of calculating a determinant using cofactor expansion, a submatrix arises for each element of the row or column you choose to expand.

For example, when computing the determinant of a 4x4 matrix, removing one row and one column forms a 3x3 submatrix. These smaller matrices are critical in calculating the determinant as they simplify the process with their more manageable size. Each submatrix contributes to the overall determinant calculation through the process of cofactor expansion.

The submatrix concept is fundamental to understanding cofactor expansion. Without recognizing the role of submatrices, breaking down the determinant calculation for larger matrices would be much more complicated. It helps to experience and simplify the large structure into understandable and workable parts, making calculations comprehensively manageable.

3x3 Determinant

The determinant of a 3x3 matrix is calculated using a specific formula. This formula is essential in matrix algebra, particularly because it serves as the basis for cofactor expansion in larger matrices.

For a 3x3 matrix:\[A = \begin{bmatrix}a & b & c \d & e & f \g & h & i \end{bmatrix}\]The determinant of matrix \( A \) is calculated as:\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

This formula breaks the determinant calculation into a series of straightforward arithmetic operations. Each term combines a main diagonal multiplication with its corresponding minor diagonal. Subtracting and adding these products according to their respective signs determines the final determinant value.

Understanding the 3x3 determinant calculation lays the foundation for solving higher order matrices' determinants. It gives insight into how determinants interact with the elements of a matrix, ultimately leading to a comprehensive understanding of the matrix's properties.