Problem 3 - Algebra for College Students

Question

For Problems \(1-18\), find the multiplicative inverse (if one exists) of each matrix. $$ \left[\begin{array}{ll} 3 & 8 \\ 2 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix is \(\begin{bmatrix} -5 & 8 \\ 2 & -3 \end{bmatrix}\).

Step 1: Determine if the Inverse Exists

To find the inverse of a matrix, the matrix must be square and have a non-zero determinant. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\). Substitute the values from the matrix: \(3 \cdot 5 - 8 \cdot 2 = 15 - 16 = -1\). Since the determinant is non-zero, an inverse exists.

Step 2: Use the Formula for the Inverse of a 2x2 Matrix

The formula for the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is \(\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Given the determinant is \(-1\), the formula becomes \(-1\begin{bmatrix} 5 & -8 \ -2 & 3 \end{bmatrix}\).

Step 3: Calculate the Inverse Matrix

Multiply each element of the matrix \(\begin{bmatrix} 5 & -8 \ -2 & 3 \end{bmatrix}\) by \(-1\). This results in: \[\begin{bmatrix} -5 & 8 \ 2 & -3 \end{bmatrix}\]. This is the multiplicative inverse of the original matrix.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix},\]this number is found using the formula \( ad - bc \). This formula helps us determine several things about the matrix, like whether an inverse exists or not.
To calculate the determinant:

• Multiply the top-left element \(a\) by the bottom-right element \(d\).
• Multiply the top-right element \(b\) by the bottom-left element \(c\).
• Subtract the second product from the first: \( ad - bc \).

For example, if a matrix is given by\[\begin{bmatrix} 3 & 8 \ 2 & 5 \end{bmatrix},\] you calculate \(3 \cdot 5 - 8 \cdot 2 = 15 - 16 = -1\).
Because the determinant is \(-1\), which is not zero, we can find an inverse for the matrix.

2x2 Matrices

2x2 matrices are the simplest square matrices you'll encounter in linear algebra. They consist of four elements arranged in two rows and two columns. These matrices are particularly important because they have a straightforward method for calculating their inverses, determinable by the formula for the determinant.
Matrices allow us to perform operations similar to those in arithmetic but on a wider scale. They are useful in various fields such as physics, computer graphics, and engineering for tasks like solving systems of equations, rotations, and transformations.
For a 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix},\]an inverse exists if the determinant \(ad - bc\) is not zero.
The formula to find the inverse is \(\frac{1}{ad-bc}\left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]\).
In simple terms, if you know the matrix elements and can calculate the determinant, you can easily find the inverse.

Matrix Algebra

Matrix algebra extends traditional arithmetic to arrays of numbers. It allows us to perform addition, subtraction, multiplication, and find inverses, contributing to solving more complex mathematical problems.
Understanding the basic operations on matrices forms the foundation for more advanced concepts in mathematics. Here are a couple of key points about matrix algebra:

• Matrix Addition and Subtraction require matrices of the same dimensions and involve element-wise operations.
• Matrix Multiplication, however, needs special attention. For multiplication, the number of columns in the first matrix must equal the number of rows in the second. The resulting matrix has dimensions matching the number of rows from the first and columns from the second.
• The inverse operation is perhaps the most intriguing and vital in matrix algebra. For a matrix to have an inverse, it has to be square (same number of rows and columns) and have a non-zero determinant.

Matrix algebra is essential in fields like computer science for tasks like solving linear programming problems and performing transformations in computer graphics.