Problem 16 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.8 & -0.3 \\ 0.5 & -0.2 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse is \(\begin{bmatrix} 20 & -30 \\ 50 & -80 \end{bmatrix}\).

Step 1: Check if Determinant is Zero

To find the inverse of a matrix, first, we need to ensure that the determinant is not zero. Calculate the determinant of matrix \(A\):\[det(A) = (0.8)(-0.2) - (-0.3)(0.5) = -0.16 + 0.15 = -0.01\]Since \(det(A) eq 0\), the inverse exists.

Step 2: Use Formula for 2x2 Matrix Inverse

For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the inverse is given by:\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]

Step 3: Substitute Values into Formula

Substitute the elements of matrix \(A\) and its determinant into the inverse formula:\[A^{-1} = \frac{1}{-0.01} \begin{bmatrix} -0.2 & 0.3 \ -0.5 & 0.8 \end{bmatrix}\]This simplifies to:\[A^{-1} = -100 \begin{bmatrix} -0.2 & 0.3 \ -0.5 & 0.8 \end{bmatrix}\]

Step 4: Calculate the Inverse Matrix

Multiply each element by \(-100\): \[A^{-1} = \begin{bmatrix} 20 & -30 \ 50 & -80 \end{bmatrix}\]So, the inverse of matrix \(A\) is:\(\begin{bmatrix} 20 & -30 \ 50 & -80 \end{bmatrix}\).

Key Concepts

Determinant Calculation

Before finding the inverse of a matrix, calculating its determinant is crucial. The determinant indicates if an inverse exists. For a \(2\times2\) matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\] the determinant is computed as \[det(A) = ad - bc.\] This number must be non-zero for the matrix to have an inverse.

In the given problem, we have a matrix \[A = \begin{bmatrix} 0.8 & -0.3 \ 0.5 & -0.2 \end{bmatrix}.\] We compute its determinant as follows:

• Multiply the diagonal elements: \( (0.8)(-0.2) = -0.16 \)
• Multiply the anti-diagonal elements: \( (-0.3)(0.5) = -0.15 \)
• Subtract the results: \( -0.16 + 0.15 = -0.01 \)

Since \(det(A) = -0.01 eq 0\), the inverse can be found.

2x2 Matrix Inverse Formula

To find the inverse of a \(2\times2\) matrix, once the determinant is non-zero, we can use a specific formula. The formula for the inverse of a given matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\] is \[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\] This formula rearranges the elements of the original matrix and changes their signs accordingly.

It's important to remember the formula modifies as follows:

• Swap the positions of \(a\) and \(d\).
• Change the signs of \(b\) and \(c\).
• Divide the entire matrix by the determinant \((ad - bc)\).

Inverse of a Matrix

Finding the inverse of a matrix is a fundamental concept in matrix algebra. The inverse \(A^{-1}\) of a matrix \(A\) is the matrix that, when multiplied with \(A\), yields the identity matrix. For a \(2\times2\) matrix, this involves checking if the matrix is invertible by ensuring the determinant is non-zero, then applying the inverse formula.

Once you have the inverse, you can use it to solve systems of equations, find matrix powers, and more. It's essential for various applications in engineering, physics, and computer science. For the given problem, after confirming a non-zero determinant, the inverse of matrix \[A = \begin{bmatrix} 0.8 & -0.3 \ 0.5 & -0.2 \end{bmatrix}\]was found to be:\[A^{-1} = \begin{bmatrix} 20 & -30 \ 50 & -80 \end{bmatrix}.\] This satisfies the condition \(A \cdot A^{-1} = I\) where \(I\) is the identity matrix.

Matrix Algebra

Matrix algebra involves operations such as addition, subtraction, and multiplication on matrices, as well as finding inverses and determinants. Understanding how to manipulate matrices is crucial for solving linear algebra problems efficiently.

Key concepts include:

• Matrix Addition: Add corresponding elements.
• Matrix Subtraction: Subtract corresponding elements.
• Matrix Multiplication: Multiply the rows of the first matrix by the columns of the second matrix.
• Inverse: Find a matrix that when multiplied with the original gives the identity matrix.

For a \(2\times2\) matrix, the concepts are simplified, and using the formula makes finding inverses relatively straightforward. Proficiency in matrix algebra enables tackling more complex problems, such as solving systems of linear equations using matrices.