Problem 15 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.6 & 0.2 \\ 0.5 & 0.1 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix} -2.5 & 5 \\ 12.5 & -15 \end{bmatrix} \).

Step 1: Check if the Inverse Exists

To determine if the inverse of a 2x2 matrix \( A \) exists, we need to compute the determinant of \( A \). The matrix \( A \) is given as \( A = \begin{bmatrix} 0.6 & 0.2 \ 0.5 & 0.1 \end{bmatrix} \). The determinant \( \det(A) \) of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). Thus, \( \det(A) = (0.6)(0.1) - (0.2)(0.5) = 0.06 - 0.10 = -0.04 \). Since the determinant is non-zero, the inverse exists.

Step 2: Apply the Formula for the Inverse

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) can be calculated using the formula \( \frac{1}{\det(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Here, \( a = 0.6 \), \( b = 0.2 \), \( c = 0.5 \), and \( d = 0.1 \). Substitute these values into the formula:\[ A^{-1} = \frac{1}{-0.04} \begin{bmatrix} 0.1 & -0.2 \ -0.5 & 0.6 \end{bmatrix} \].

Step 3: Calculate the Multiplicative Factors

Calculate the factor \( \frac{1}{-0.04} \) which simplifies to \( -25 \). This is because dividing by a negative fraction is equivalent to multiplying by its reciprocal. Thus, \( \frac{1}{-0.04} = -25 \).

Step 4: Multiply to Find the Inverse

Multiply the matrix by the factor calculated in the previous step:\[A^{-1} = -25 \begin{bmatrix} 0.1 & -0.2 \ -0.5 & 0.6 \end{bmatrix} = \begin{bmatrix} -2.5 & 5 \ 12.5 & -15 \end{bmatrix} \].This is done by multiplying each element of the matrix by \(-25\).

Key Concepts

2x2 Matrices

A 2x2 matrix is a simple rectangular array consisting of two rows and two columns. These matrices are commonly used in linear algebra and have the general form:

• Row 1: [a, b]
• Row 2: [c, d]

The entries (a, b, c, and d) can be any numbers—integers, fractions, or decimals. Understanding this basic structure is crucial because it forms the foundation for performing operations like matrix inversion.
Every element in a 2x2 matrix plays a distinct role in calculations such as finding the determinant or the inverse. This understanding allows for precise manipulation and extraction of necessary elements during mathematical operations.

Determinant Calculation

Calculating the determinant of a 2x2 matrix is a fundamental step in determining whether an inverse exists. For a matrix of the form:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]The determinant is computed as:\[ad - bc\]This tells us the area scaling factor of the transformation indicated by the matrix. If the determinant is zero, the matrix is singular, meaning it does not have an inverse. In our example matrix:\[A = \begin{bmatrix} 0.6 & 0.2 \ 0.5 & 0.1 \end{bmatrix}\]The determinant is calculated as follows:

• Multiply diagonal elements: \(0.6 \times 0.1 = 0.06\)
• Multiply off-diagonal elements: \(0.2 \times 0.5 = 0.10\)
• Subtract the results: \(0.06 - 0.10 = -0.04\)

Since the determinant is \(-0.04\), which is not zero, the inverse exists.

Inverse Matrix Formula

Once you've confirmed that the inverse exists by having a non-zero determinant, you can apply the inverse matrix formula.For our 2x2 matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]The formula to find the inverse is:\[\frac{1}{ad - bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]In our specific example:

• The matrix is \(A = \begin{bmatrix} 0.6 & 0.2 \ 0.5 & 0.1 \end{bmatrix}\)
• The determinant is \(-0.04\)
• By the formula: \(A^{-1} = \frac{1}{-0.04} \begin{bmatrix} 0.1 & -0.2 \ -0.5 & 0.6 \end{bmatrix}\)
• Calculate \(\frac{1}{-0.04}\) to find it equals \(-25\)
• Now multiply each element by \(-25\) to get: \(\begin{bmatrix} -2.5 & 5 \ 12.5 & -15 \end{bmatrix}\)

This final matrix \(\begin{bmatrix} -2.5 & 5 \ 12.5 & -15 \end{bmatrix}\) is the inverse. Any further matrix operations, such as verifying the result by multiplying with the original matrix, will show how these components cancel to produce an identity matrix.