Problem 14 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} 5 & 10 \\ -3 & -6 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \( A \) does not have an inverse because its determinant is 0.

Step 1: Verify Matrix is 2x2

Confirm that the matrix \( A \) is 2x2. This matrix is \(A=\begin{bmatrix} 5 & 10 \ -3 & -6 \end{bmatrix}\), which is a 2x2 matrix.

Step 2: Calculate the Determinant of A

The determinant for a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( \text{det}(A) = ad - bc \). For our matrix, \( a = 5 \), \( b = 10 \), \( c = -3 \), and \( d = -6 \). Substitute the values into the formula:\[\text{det}(A) = (5)(-6) - (10)(-3) = -30 + 30 = 0\]The determinant is 0.

Step 3: Determine if Inverse Exists

A matrix has an inverse if and only if its determinant is non-zero. Here, since \( \text{det}(A) = 0 \), the matrix \( A \) does not have an inverse.

Key Concepts

Determinant

In the study of matrices, the determinant is a special number that gives us significant insights into a matrix's properties. For a 2x2 matrix, the determinant can help us determine if the matrix is invertible. The formula to calculate the determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is:\[\text{det}(A) = ad - bc\]This formula involves multiplying the elements on the main diagonal \( (a \times d) \) and subtracting the product of the elements on the other diagonal \( (b \times c) \).
The determinant is crucial because:

• If the determinant is not equal to zero, the matrix can potentially have an inverse.
• If the determinant is zero, the matrix does not have an inverse. This is because the matrix is considered "singular" and doesn't have a unique solution for its inversion.

When working with 2x2 matrices, always calculate the determinant first to guide your next steps.

2x2 Matrix

A 2x2 matrix is a rectangular array of numbers with two rows and two columns. It is a simple yet powerful structure often used in linear algebra to solve systems of equations, perform coordinate transformations, and more. In our exercise, we examined the matrix:\[A = \begin{bmatrix} 5 & 10 \ -3 & -6 \end{bmatrix}\]Each element of the matrix is identified by its position.
The matrix is organized as follows:

• \( a = 5 \)
• \( b = 10 \)
• \( c = -3 \)
• \( d = -6 \)

Understanding the structure and notation of a 2x2 matrix is crucial for performing operations like finding determinants and checking for invertibility.
Pay attention to the order of rows and columns, as mixing them up can lead to incorrect calculations.

Inverse Matrix

An inverse matrix, denoted as \( A^{-1} \), is a matrix that, when multiplied by the original matrix \( A \), results in the identity matrix. The identity matrix for a 2x2 matrix looks like this:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]For a matrix to have an inverse, its determinant must be non-zero. In the provided exercise, the determinant was zero, which means the matrix \( A \) does not have an inverse. But when a 2x2 matrix has a non-zero determinant, the inverse can be found using the following formula:\[A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]This formula swaps \( a \) and \( d \), negates \( b \) and \( c \), then divides each element by the determinant.
Without a non-zero determinant, this process becomes impossible as division by zero is undefined.
Remember, not all matrices have inverses, and checking the determinant is the first step in the process!