Problem 22 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} 1 & 2 & 5 \\ -2 & -3 & 2 \\ 3 & 5 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has no inverse because its determinant is zero.

Step 1: Write Down the Matrix

We have the matrix:\[\begin{bmatrix}1 & 2 & 5 \-2 & -3 & 2 \3 & 5 & 3 \end{bmatrix}\] Our goal is to find its determinant.

Step 2: Calculate First Row Cofactors

The determinant of a 3x3 matrix can be calculated using the rule of Sarrus or the cofactor expansion along a row or column. We will use cofactor expansion along the first row, which gives us three products:\[ det(A) = 1 \cdot \det\begin{bmatrix} -3 & 2 \ 5 & 3 \end{bmatrix} - 2 \cdot \det\begin{bmatrix} -2 & 2 \ 3 & 3 \end{bmatrix} + 5 \cdot \det\begin{bmatrix} -2 & -3 \ 3 & 5 \end{bmatrix} \]

Step 3: Calculate 2x2 Determinants

Calculate the determinants of the 2x2 matrices:- \( \det\begin{bmatrix} -3 & 2 \ 5 & 3 \end{bmatrix} = (-3)(3) - (2)(5) = -9 - 10 = -19 \)- \( \det\begin{bmatrix} -2 & 2 \ 3 & 3 \end{bmatrix} = (-2)(3) - (2)(3) = -6 - 6 = -12 \)- \( \det\begin{bmatrix} -2 & -3 \ 3 & 5 \end{bmatrix} = (-2)(5) - (-3)(3) = -10 + 9 = -1 \)

Step 4: Substitute and Simplify

Substitute these values back into the cofactor expansion formula:\[ det(A) = 1(-19) - 2(-12) + 5(-1) \]Simplifying this gives:\[ det(A) = -19 + 24 - 5 \]\[ det(A) = 0 \]

Step 5: Determine Invertibility

A matrix is invertible if its determinant is non-zero. Here, the determinant of the matrix is 0, which means the matrix does not have an inverse.

Key Concepts

Matrix Inverse

The matrix inverse is a fundamental concept in linear algebra. If you multiply a matrix by its inverse, the result is the identity matrix.
The identity matrix is like the number 1 for matrices, maintaining the properties of the original matrix.

• If a matrix has an inverse, it is called invertible or non-singular.
• Otherwise, it is said to be non-invertible or singular.

The presence or absence of an inverse depends on the determinant.
An invertible matrix has a non-zero determinant, while a singular matrix has a determinant equal to zero. Knowing whether a matrix can be inverted is essential in solving linear equations, amongst other applications in engineering and computer science.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a technique for calculating the determinant of a matrix. This method involves expanding the determinant across a chosen row or column.
Let's go through the main steps:

• Select a row or column. A common choice is the first row owing to its straightforwardness.
• For each element in this row or column, calculate its cofactor. The cofactor is obtained by covering the current row and column to produce a smaller matrix, known as the minor, and then finding its determinant.
• Multiply each element by its corresponding cofactor.
• Sum these products to obtain the determinant.

This approach leverages smaller matrices (2x2 for a 3x3 case) to efficiently and systematically calculate the larger matrix's determinant. It is versatile and can be used for any size matrix, though computationally it becomes heavier with larger matrices.

3x3 Matrix Determinant

Determining the determinant of a 3x3 matrix is a bit more involved than a 2x2 matrix, but it offers a deeper understanding of matrix properties.
The determinant helps in assessing properties like invertibility. For a 3x3 matrix, we commonly use the method of cofactor expansion. Let's outline the key process:

• The first step is to select a row or column. It's common to start with the first row for simplicity.
• Calculate the determinant of each 2x2 minor, the submatrix formed by removing the current element's row and column.
• Multiply each element by its minor's determinant and an alternating sign, based on their position (-1 raised to the sum of the row index and column index).
• Sum all these calculated values to get the 3x3 matrix's determinant.

A zero determinant indicates that the matrix is singular, lacking an inverse. This concept is crucial for various applications in physics, mathematics, and even economics where systems of linear equations are solved using matrices.