Problem 12 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The inverse of matrix \(A\) is \( \begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{bmatrix} \).

Step 1: Compute the Determinant

First, calculate the determinant of matrix \(A\). The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \(ad - bc\). For matrix \( A = \begin{bmatrix} -1 & 2 \ -2 & -1 \end{bmatrix} \), the determinant is \((-1)(-1) - (2)(-2) = 1 - (-4) = 5\).

Step 2: Check Invertibility

A matrix is invertible only if its determinant is non-zero. Since the determinant of matrix \(A\) is \(5\), which is non-zero, matrix \(A\) is invertible, and hence, we can find \(A^{-1}\).

Step 3: Use the Inverse Formula

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), if the determinant is non-zero, the inverse is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). For matrix \( A = \begin{bmatrix} -1 & 2 \ -2 & -1 \end{bmatrix} \), this becomes \( \frac{1}{5} \begin{bmatrix} -1 & -2 \ 2 & -1 \end{bmatrix} \).

Step 4: Simplify the Inverse

Multiply each element of the matrix \( \begin{bmatrix} -1 & -2 \ 2 & -1 \end{bmatrix} \) by \( \frac{1}{5} \). This gives \( \begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \ \frac{2}{5} & -\frac{1}{5} \end{bmatrix} \).

Key Concepts

Determinant Calculation

Before determining if a matrix is invertible, you need to calculate its determinant. The determinant is a special number that helps in understanding the matrix properties like invertibility. For a 2x2 matrix, such as:

• \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

The determinant is calculated using the formula: \(ad - bc\).
To visualize, if each element of the matrix is like a part of a simple machine, the determinant tells you if the machine works correctly.
For example, given matrix \(A\):

• \[\begin{bmatrix} -1 & 2 \ -2 & -1 \end{bmatrix}\]

You calculate the determinant as \((-1) \times (-1) - (2) \times (-2)\).
This results in \(1 - (-4) = 5\).
Calculating this correctly is key to finding if the matrix can be inverted.

Invertibility Condition

A matrix's invertibility depends directly on its determinant. This is like the "golden ticket" for a matrix to have an inverse.
In essence, a matrix is invertible if its determinant is not zero. Imagine it like trying to perform a magic trick. Without the right magic number, the trick just won’t work!
For our matrix \(A\), we found that its determinant is \(5\).
Since \(5\) is greater than zero, it confirms that our matrix is indeed invertible. This opens the door to actually finding an inverse matrix, as the presence of a non-zero determinant guarantees the process will succeed.
Always ensure your calculated determinant is accurate before proceeding to find an inverse.

Inverse Formula for 2x2 Matrix

Finding an inverse is straightforward once you confirm a matrix is invertible. There’s a neat formula for a 2x2 matrix, which makes it all quite simple.
Once you've confirmed invertibility, here's the formula for a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\):

• \[\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]

This formula swaps the diagonal elements and changes the signs of the off-diagonal ones, and scales the entire matrix by the reciprocal of its determinant.
Applying this to matrix \(A\), whose determinant we calculated as 5:

• The inverse matrix is:\[\frac{1}{5}\begin{bmatrix} -1 & -2 \ 2 & -1 \end{bmatrix}\]

After simplification, this results in the matrix:

• \[\begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \ \frac{2}{5} & -\frac{1}{5} \end{bmatrix}\]

Understanding the flow from determinant calculation to applying the formula helps solidify the process of matrix inversion.