Problem 9 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 3 & 7 \\ 2 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix} 5 & -7 \\ -2 & 3 \end{bmatrix} \).

Step 1: Understand Conditions for Inversibility

A matrix \(A\) is invertible if and only if its determinant \( \det(A) eq 0 \). For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is given by \( ad - bc \).

Step 2: Calculate the Determinant

For the matrix \( A = \begin{bmatrix} 3 & 7 \ 2 & 5 \end{bmatrix} \), the determinant is \(3 \cdot 5 - 7 \cdot 2 = 15 - 14 = 1 \). Since the determinant is non-zero, the matrix is invertible.

Step 3: Apply the Formula for the Inverse of a 2x2 Matrix

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).

Step 4: Substitute Values into the Inverse Formula

Substitute the values from matrix \(A\): \\[ A^{-1} = \frac{1}{1} \begin{bmatrix} 5 & -7 \ -2 & 3 \end{bmatrix} = \begin{bmatrix} 5 & -7 \ -2 & 3 \end{bmatrix} \] \Thus, the inverse of matrix \(A\) is \( \begin{bmatrix} 5 & -7 \ -2 & 3 \end{bmatrix} \).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, like the one used in the given exercise, the determinant helps us understand important properties about the matrix, such as whether it can be inverted or not.
For a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \). This formula multiplies the elements diagonally and then subtracts the product of the other diagonal.
In simpler terms, if you imagine weaving through the numbers from top-left to bottom-right and from top-right to bottom-left, the determinant gives you a numerical understanding of the matrix's "leveraging power."

• A determinant of zero often means the matrix doesn't have an inverse.
• If the determinant is non-zero, like in our exercise where it's \( 1 \), the matrix can indeed be inverted.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is a matrix that has an inverse.
The inverse of a matrix \( A \), denoted as \( A^{-1} \), when multiplied back with the original matrix \( A \), results in the identity matrix. This means the product \( AA^{-1} = I \), where \( I \) is the identity matrix characterized by 1s on its diagonal and 0s elsewhere.

• Only matrices with a non-zero determinant have an inverse.
• The concept of an inverse is crucial in solving linear equations and in various applications across physics, engineering, and computer science.

In our exercise, since the determinant of matrix \( A \) is 1, it is invertible, allowing us to compute \( A^{-1} \) using the inverse formula.

2x2 Matrix

A 2x2 matrix is the simplest form of a square matrix, containing two rows and two columns. Its layout is often expressed as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
Despite its simplicity, a 2x2 matrix is quite powerful for a variety of calculations. It can represent systems of linear equations and transformations in two-dimensional space such as rotations and scaling.
The operations that you perform with 2x2 matrices are foundational to more complex matrix operations performed in higher dimensions:

• Matrix addition and subtraction.
• Matrix multiplication.
• Finding the determinant and the inverse.

Being proficient with 2x2 matrices enhances understanding when dealing with larger matrices.

Matrix Algebra

Matrix algebra is the branch of mathematics that studies matrices and their properties. It expands upon traditional algebra by introducing operations that are possible on matrices such as addition, subtraction, and most importantly, multiplication.
Matrix multiplication is not as straightforward as it is with regular numbers. For instance, the product of two matrices depends on the order in which they are multiplied, meaning \( AB \) might not necessarily equal \( BA \).

• The concept of the inverse, as applied in the exercise, is an example of matrix algebra in action.
• Understanding the multiplication of matrices and finding inverses is essential for solving systems of equations, among many other applications.

By practicing matrix algebra with smaller matrices like 2x2, you can build the intuitive understanding required for larger matrix systems.