Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{cc} -4 & -3 \\ -5 & 8 \end{array}\right] $$

Step-by-Step Solution

Answer

The multiplicative inverse is \( \begin{bmatrix} -\frac{8}{47} & -\frac{3}{47} \\ -\frac{5}{47} & \frac{4}{47} \end{bmatrix} \).

Step 1: Check if the matrix is invertible

First, we need to check if the given matrix is invertible by calculating its determinant. The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). Here, the matrix is \( \begin{bmatrix} -4 & -3 \ -5 & 8 \end{bmatrix} \), so we calculate the determinant as follows: \( (-4)(8) - (-3)(-5) = -32 - 15 = -47 \). Since the determinant is not zero, the matrix is invertible.

Step 2: Use the formula for the inverse of a 2x2 matrix

Since the determinant is non-zero, we can find the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) using the formula: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]. For our matrix \( \begin{bmatrix} -4 & -3 \ -5 & 8 \end{bmatrix} \), this becomes: \[ \frac{1}{-47} \begin{bmatrix} 8 & 3 \ 5 & -4 \end{bmatrix} \].

Step 3: Simplify the inverse matrix

Now, we apply the scalar multiplication for each element of the matrix: \[ \begin{bmatrix} 8 & 3 \ 5 & -4 \end{bmatrix} \times \left(\frac{1}{-47}\right) \]. Calculating each term separately, we get: \[ \begin{bmatrix} \frac{8}{-47} & \frac{3}{-47} \ \frac{5}{-47} & \frac{-4}{-47} \end{bmatrix} = \begin{bmatrix} -\frac{8}{47} & -\frac{3}{47} \ -\frac{5}{47} & \frac{4}{47} \end{bmatrix} \]. This is the inverse matrix.

Key Concepts

Determinant of a 2x2 Matrix

To determine if a 2x2 matrix is invertible, we first need to calculate its determinant. The determinant of a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is found using the formula \( ad - bc \). A non-zero determinant indicates that the matrix is invertible, while a zero determinant means it is not.

Calculating the determinant involves multiplying the diagonal elements and then subtracting the product of the off-diagonal elements. For our given matrix \( \begin{bmatrix} -4 & -3 \ -5 & 8 \end{bmatrix} \):

• Multiply \( -4 \times 8 = -32 \)
• Multiply \( -3 \times -5 = 15 \)
• Subtract these results: \( -32 - 15 = -47 \)

Since the determinant \(-47\) is not zero, the matrix can be inverted.

Scalar Multiplication

Scalar multiplication in the context of matrices refers to multiplying every element of the matrix by a scalar (a single number). In our case, the scalar comes from the inverse matrix formula \( \frac{1}{ad-bc} \). This scalar is necessary to apply after adjusting the layout of the original matrix elements to obtain the inverse.

For the matrix \( \begin{bmatrix} -4 & -3 \ -5 & 8 \end{bmatrix} \), once it's rearranged to \( \begin{bmatrix} 8 & 3 \ 5 & -4 \end{bmatrix} \), the determinant \(-47\) was used as the scalar. By multiplying all elements by \( \frac{1}{-47} \), we get:

• \( \frac{8}{-47} \)
• \( \frac{3}{-47} \)
• \( \frac{5}{-47} \)
• \( \frac{-4}{-47} \)

We arrive at the final inverse matrix, ensuring that the matrix maintains its structure under this scalar transformation.

Invertible Matrix

An invertible matrix, or a non-singular matrix, is one that can be reversed to find its original form by using its inverse. For a 2x2 matrix, the primary condition for invertibility is a non-zero determinant.

If a matrix is invertible, it implies that it has an inverse matrix that, when multiplied with the original, results in the identity matrix, \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). This principle is crucial in solving system of linear equations, where an invertible matrix allows us to simplify complex problems efficiently.

When checking for invertibility of our matrix \( \begin{bmatrix} -4 & -3 \ -5 & 8 \end{bmatrix} \), and finding that the determinant is \(-47\), confirms its potential to yield an inverse.

Matrix Algebra

Matrix algebra is the fundamental framework that allows complex calculations involving matrices to be handled systematically. It encompasses operations such as addition, subtraction, multiplication, and finding inverses, among others. Using these operations, we can solve various mathematical and real-world problems efficiently.

The development of matrix algebra provides powerful tools for enhancing computational efficiency in fields such as physics, economics, computer science, and engineering. In this exercise, matrix algebra has shown how an inventory of operations can be systematically applied to derive the inverse of a matrix.

Understanding these principles and their interplay, particularly how transformations applied to a matrix lead toward finding its inverse, is fundamental to leveraging matrix algebra effectively in problem solving.