Problem 14 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}-7 & 4 \\\8 & -5\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse is \(\begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \\ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix}\).

Step 1: Understanding the Conditions

In order to find the inverse of a 2x2 matrix \(\begin{bmatrix}a & b \c & d\end{bmatrix}\), the determinant must not be zero. The formula for the determinant is \(ad - bc\). If the determinant is zero, the matrix does not have an inverse.

Step 2: Compute the Determinant

Calculate the determinant of the matrix \(\begin{bmatrix}-7 & 4 \8 & -5\end{bmatrix}\). Using the determinant formula: \((-7)(-5) - (8)(4)\), you get \(35 - 32 = 3\). Since the determinant is not zero, the matrix has an inverse.

Step 3: Apply the Formula for Inverse

The inverse of a 2x2 matrix \(\begin{bmatrix}a & b \c & d\end{bmatrix}\) is \(\frac{1}{ad-bc}\begin{bmatrix}d & -b \-c & a\end{bmatrix}\). Substitute the values: \(\frac{1}{3}\begin{bmatrix}-5 & -4 \-8 & -7\end{bmatrix}\).

Step 4: Simplifying the Expression

Distribute \(\frac{1}{3}\) across the matrix to get the inverse: \[\begin{bmatrix}\frac{-5}{3} & \frac{-4}{3}\\frac{-8}{3} & \frac{-7}{3}\end{bmatrix}\]This matrix is the inverse of the original matrix.

Key Concepts

Matrix Determinant

The concept of a matrix determinant is a crucial part of understanding matrix algebra, especially when working with inverse matrices. The determinant is a scalar value that can be calculated from a square matrix and provides important properties of the matrix. For a 2x2 matrix, the determinant is found using the formula:
\[\text{det}(A) = ad - bc\]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix \(\begin{bmatrix}a & b \ c & d\end{bmatrix}\).
Understanding the determinant is essential as it helps to determine invertibility. If the determinant is zero, it indicates that the matrix does not have an inverse. This is because a determinant of zero means that the matrix transforms space to a lower dimension, or, in simpler terms, the rows or columns are linearly dependent; thus, it's not invertible.

2x2 Matrix

A 2x2 matrix is a simple, yet highly instructive illustration used in matrix algebra. It consists of two rows and two columns, making it one of the smallest and easiest square matrices to work with. The general form of a 2x2 matrix is:
\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]Even though 2x2 matrices are small, they serve as the basis for understanding larger matrices and more complex operations. Operations like addition, subtraction, multiplication, as well as finding the determinant and inverse, often start with such matrices. They are simple enough to handle manually, making them perfect for learning and teaching purposes.
When dealing with inverses, as seen in exercises with 2x2 matrices, we begin by calculating the determinant. If non-zero, we proceed to find the inverse using specific formulas. This makes mastering 2x2 matrices fundamental before moving on to understand higher-dimensional matrices.

Matrix Algebra

Matrix algebra is a powerful mathematical tool that extends basic algebra to matrices, allowing us to perform complex calculations on arrays of numbers. It encompasses various operations, including addition, subtraction, scalar multiplication, and matrix multiplication. Importantly, it also includes determining determinants and finding inverses, as illustrated in our exercise.
Key skills in matrix algebra involve:

• Understanding matrix dimensions and how they interact during multiplication.
• Mastering the use of determinants to check for matrix invertibility.
• Utilizing matrix inverses to solve systems of equations or transform matrices.

This branch of mathematics is essential in fields such as physics, computer science, engineering, and statistics, where systems of equations are frequently encountered. Grasping the basics of matrix algebra with examples like finding the inverse of a 2x2 matrix provides a solid foundation for more advanced studies.