Problem 17 - College Algebra with Modeling and Visualization

Question

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 3 & -1 & 2 \\ 0 & 5 & 7 \\ 1 & 0 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of matrix \(A\) is \(-32\).

Step 1: Understand the Determinant Formula

The determinant of a 3x3 matrix \(A\) with elements \(a_{ij}\) can be found using cofactor expansion along any row or column. We will expand along the first row. The formula is: \( \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \), where \(C_{ij}\) is the cofactor of \(a_{ij}\).

Step 2: Calculate Cofactor \(C_{11}\)

Remove the first row and first column. The minor is the 2x2 matrix \(\left[\begin{array}{cc} 5 & 7 \ 0 & -1 \end{array}\right]\). The determinant of this matrix is: \(5(-1) - 0(7) = -5\). The cofactor \(C_{11} = (-1)^{1+1} \times (-5) = -5\).

Step 3: Calculate Cofactor \(C_{12}\)

Remove the first row and second column. The minor is the 2x2 matrix \(\left[\begin{array}{cc} 0 & 7 \ 1 & -1 \end{array}\right]\). The determinant of this matrix is: \(0(-1) - 7(1) = -7\). The cofactor \(C_{12} = (-1)^{1+2} \times (-7) = 7\).

Step 4: Calculate Cofactor \(C_{13}\)

Remove the first row and third column. The minor is the 2x2 matrix \(\left[\begin{array}{cc} 0 & 5 \ 1 & 0 \end{array}\right]\). The determinant of this matrix is: \(0(0) - 5(1) = -5\). The cofactor \(C_{13} = (-1)^{1+3} \times (-5) = -5\).

Step 5: Compute the Determinant

Using the cofactors and the formula from Step 1, calculate: \( \text{det}(A) = 3(-5) - 1(7) + 2(-5) = -15 - 7 - 10 = -32 \).

Key Concepts

Cofactor Expansion

The cofactor expansion method is a powerful tool for calculating determinants, particularly useful for 3x3 matrices. This approach involves expanding the determinant along a chosen row or column. In this context, a cofactor (\(C_{ij}\)) is calculated by removing the row and column of a specific element (\(a_{ij}\)) of the matrix and finding the determinant of the resulting smaller matrix, known as a minor.

• The formula used for cofactor expansion is: \[\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}\]where \(a_{11}, a_{12},\) and \(a_{13}\) are elements of the first row and their respective cofactors are \(C_{11}, C_{12},\) and \(C_{13}\).
• In this method, each element's cofactor is determined by multiplying the minor's determinant by \((-1)^{i+j}\) to account for the correct sign.

Although this may seem complex initially, practice helps make this method straightforward and reliable for computing determinants in matrix algebra.

3x3 Matrix Determinant

The calculation of a 3x3 matrix determinant is a fundamental concept in matrix algebra.

• A determinant is a special number that can be computed from a square matrix. It provides information about the matrix, such as its invertibility, among other properties.
• For a 3x3 matrix, using the cofactor expansion method simplifies the computation by focusing on smaller 2x2 matrices, or minors.
• The resultant determinant helps in various applications like solving systems of linear equations, computing inverses of matrices, and more.

Understanding how to calculate determinants is essential for advancing in mathematics and engineering fields.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrix operations. It is utilized across sciences and engineering.

• Key operations include matrix addition, multiplication, inversion, and finding determinants. Each operation follows specific rules and properties.
• Determinants, in particular, are critical in matrix algebra as they reveal essential characteristics of matrices. For example, a non-zero determinant indicates that a matrix is invertible, while a zero determinant does not.
• Matrix algebra forms the basis for many advanced topics, such as linear transformations, eigenvalues, and linear equations.

Mastering these basics allows for solving complex problems and understanding deeper mathematical concepts.