Problem 12 - A Graphical Approach to College Algebra

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: Check if the Inverse Exists

To determine if the matrix \(A\) has an inverse, we need to calculate its determinant. The matrix is \(A = \begin{bmatrix} -1 & 2 \ -2 & -1 \end{bmatrix}\). The determinant, \(\det(A)\), is calculated as: \(-1\times(-1) - (2)\times(-2) = 1 - (-4) = 1 + 4 = 5\). Since the determinant is not zero, the inverse exists.

Step 2: Use the Formula for the Inverse

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), if the determinant is not zero, the inverse is given by: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]. In our case, \(a = -1\), \(b = 2\), \(c = -2\), and \(d = -1\). Substituting, we get: \[ A^{-1} = \frac{1}{5} \begin{bmatrix} -1 & -2 \ 2 & -1 \end{bmatrix} \].

Step 3: Simplify the Inverse Matrix

Now, we simply scale the matrix by \(\frac{1}{5}\). Therefore, \[ A^{-1} = \begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \ \frac{2}{5} & -\frac{1}{5} \end{bmatrix} \]. This is the inverse matrix \(A^{-1}\).

Key Concepts

Determinant Calculation

Calculating the determinant is a crucial step to find out if a matrix has an inverse. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is computed using the formula: \( ad - bc \). This determinant value helps determine if a matrix is invertible.

In our example, the matrix \( A = \begin{bmatrix} -1 & 2 \ -2 & -1 \end{bmatrix} \) requires calculating the determinant as follows:

• Multiply \( -1 \) by \( -1 \), which gives \( 1 \).
• Multiply \( 2 \) by \( -2 \), which results in \( -4 \).
• Add these results: \( 1 - (-4) = 1 + 4 = 5 \).

The result, \( 5 \), is non-zero. This indicates that an inverse exists. A non-zero determinant shows that the matrix can be reliably reversed into another matrix, thus confirming the existence of an inverse.

2x2 Matrix

A 2x2 matrix is a simple form of a matrix with two rows and two columns, commonly written as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). It is one of the most basic matrix structures imported and used in various mathematical fields, in particular, linear algebra.

The elements of a 2x2 matrix are referred to as follows:

• \( a \) is the element in the first row, first column.
• \( b \) is the element in the first row, second column.
• \( c \) is the element in the second row, first column.
• \( d \) is the element in the second row, second column.

This simple structure makes calculations straightforward, such as multiplication, addition, and finding determinants, as opposed to larger matrices.

Inverse Matrix Formula

The inverse of a matrix is a key concept in linear algebra. It essentially reverses the effect of the original matrix. For a 2x2 matrix, provided the determinant is not zero, the formula for finding the inverse is given by:\[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]

In this formula,

• \( \det(A) \) is the determinant of the original matrix, as previously calculated.
• \( d, \ -b, \ -c, \ a \) are adjusted positions for elements in the original matrix.

Applying this to our matrix \( A = \begin{bmatrix} -1 & 2 \ -2 & -1 \end{bmatrix} \),we substitute the values into the formula:\( A^{-1} = \frac{1}{5} \begin{bmatrix} -1 & -2 \ 2 & -1 \end{bmatrix} \).
The resulting inverse matrix is:\( \begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \ \frac{2}{5} & -\frac{1}{5} \end{bmatrix} \),showcasing how we took each element and adjusted them based on the formula for the inverse.