Problem 7 - Precalculus: Mathematical for Calculus

Question

Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll}5 & 3 \\ 3 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

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

Step 1: Determine the Determinant

To find the inverse of a 2x2 matrix, the first step is to calculate the determinant. The formula for the determinant \( D \) of the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( D = ad - bc \). For our matrix \( \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix} \), the determinant is: \[ D = (5)(2) - (3)(3) = 10 - 9 = 1 \].

Step 2: Check Invertibility

A matrix is invertible if its determinant is not equal to zero. In our case, the determinant is 1. Since 1 is not zero, the matrix is invertible.

Step 3: Use the Formula for Inverse

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \[ \frac{1}{D} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] where \( D \) is the determinant. Substituting the values from our matrix and \( D = 1 \), the inverse is: \[ \begin{bmatrix} 2 & -3 \ -3 & 5 \end{bmatrix} \].

Key Concepts

Understanding the Determinant

The determinant is a crucial factor when working with matrices, especially when we're seeking to find an inverse of a matrix. For any 2x2 matrix, say \[\begin{bmatrix} a & b \ c & d \end{bmatrix},\]the determinant \(D\) is computed using the formula: \(D = ad - bc\).
This simple calculation results in a scalar value, which actually gives us no direct visual insight into the matrix, but it carries significant weight.

• First of all, if this value is equal to zero, the matrix is labeled as non-invertible, meaning an inverse doesn't exist.
• If the determinant is not zero, the matrix can be inverted, and this very number influences how the inverse matrix behaves.

In our example, the matrix \[\begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix}\]has a determinant of 1, implying that inversion is possible.

The Nature of a 2x2 Matrix

When working with matrices, especially in introductory linear algebra, the 2x2 matrix often serves as a stepping stone to understanding larger and more complex matrix structures. A 2x2 matrix is categorized by two rows and two columns:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]This simplicity makes 2x2 matrices an excellent learning tool.

• Firstly, they're relatively easy to visualize, making computations like determinants and inverses more tangible.
• They serve as foundational examples where concepts "just work" before scaling these methods to larger matrices.

Calculating determinants and inverses for 2x2 matrices follows a straightforward formula, making them ideal first steps when mastering basic matrix operations. This ease is exemplified in our problem where matrix inverses and determinants were computed effortlessly.

Determining an Invertible Matrix

An invertible matrix is a core concept in linear algebra, referring to a matrix that has an inverse. Imagine it as a matrix with a "counterpart" that can revert any transformation back to its original state. For a 2x2 matrix to be invertible, its determinant must not be zero.
Why does the determinant matter? Let's break it down:

• A zero determinant means the matrix squishes transformations into a lower dimension, losing information, which cannot be undone.
• If the determinant is non-zero, the transformation is stable and reversible, allowing for the "counteraction" that an inverse provides.

In the context of our problem, the determinant of the matrix \[\begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix}\]was calculated to be 1, confirming the matrix as invertible. By using the formula for the inverse of a 2x2 matrix \[\frac{1}{D} \begin{bmatrix} d & -b \ -c & a \end{bmatrix},\]we successfully identified its inverse. This showcases the influence of determinants in ascertaining the invertibility of matrices, reinforcing the crucial role played by determinants in linear transformations.