Problem 13 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}2 & 5 \\\\-5 & -13\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} 13 & 5 \\ -5 & -2 \end{bmatrix} \).

Step 1: Check if the Matrix has an Inverse

For a matrix to have an inverse, its determinant must be non-zero. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\). For the given matrix \(\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix}\), we calculate the determinant as follows:\[ \text{det} = (2)(-13) - (5)(-5) = -26 + 25 = -1 \]Since the determinant is \(-1\), which is not zero, the matrix has an inverse.

Step 2: Find the Adjugate of the Matrix

To find the inverse, we first determine the adjugate (also known as the adjoint) of the matrix. Swap the positions of \(a\) and \(d\), and change the signs of \(b\) and \(c\). The adjugate of the matrix \(\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix}\) is:\[\begin{bmatrix} -13 & -5 \ 5 & 2 \end{bmatrix}\]

Step 3: Calculate the Inverse matrix using the Determinant and Adjugate

The inverse of a matrix \(A\) is given by \(A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adjugate}(A)\). We have determined that \(\text{det}(A) = -1\) and the adjugate is \(\begin{bmatrix} -13 & -5 \ 5 & 2 \end{bmatrix}\). Thus, the inverse matrix is:\[A^{-1} = \frac{1}{-1} \cdot \begin{bmatrix} -13 & -5 \ 5 & 2 \end{bmatrix} = \begin{bmatrix} 13 & 5 \ -5 & -2 \end{bmatrix}\]

Step 4: Verify the Inverse

To ensure our solution is correct, we multiply the original matrix by its inverse and check if the result is the identity matrix:\[\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix} \begin{bmatrix} 13 & 5 \ -5 & -2 \end{bmatrix} = \begin{bmatrix} (2)(13)+(5)(-5) & (2)(5)+(5)(-2) \ (-5)(13)+(-13)(-5) & (-5)(5)+(-13)(-2) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]Since the result is the identity matrix, our inverse is correct.

Key Concepts

Determinant of a Matrix

The determinant of a matrix plays a crucial role in determining whether a matrix can be inverted. When dealing with a 2x2 matrix like \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), finding the determinant is quite straightforward. You use the formula: \[ \text{det}(A) = ad - bc \]This formula involves multiplying the diagonal elements \(a\) and \(d\), then subtracting the product of the other diagonal elements, \(b\) and \(c\).

In our example, for the matrix \(\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix}\), we find:

• First, calculate \( (2)(-13) = -26 \).
• Then calculate \( (5)(-5) = -25 \).
• The determinant is \(-26 + 25 = -1\).

A non-zero determinant such as \(-1\) confirms that the matrix has an inverse.

Adjugate of a Matrix

The adjugate of a matrix, sometimes referred to as the adjoint, is an essential component when finding the inverse of a matrix. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the adjugate is determined by rearranging the elements in a specific pattern:

• Swap the elements \(a\) and \(d\).
• Change the signs of \(b\) and \(c\).

Of course, these instructions need practical examples to make sense. Let's consider the matrix \(\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix}\):

• Swap \(2\) and \(-13\).
• Change the signs of \(5\) and \(-5\).

Thus, the adjugate results in \(\begin{bmatrix} -13 & -5 \ 5 & 2 \end{bmatrix}\). Determining the adjugate correctly is vital as it is directly used in calculating the matrix inverse.

Matrix Multiplication

Matrix multiplication is used in verifying if a calculated inverse is correct. It involves combining two matrices to form a new matrix. For 2x2 matrices, follow these steps:

• Multiply the elements of the first row of the first matrix by the corresponding elements of the column of the second matrix and sum them up.
• Repeat this process for all rows and columns.

For example, multiplying the original matrix \(\begin{bmatrix} 2 & 5 \ -5 & -13 \end{bmatrix}\) by the calculated inverse \(\begin{bmatrix} 13 & 5 \ -5 & -2 \end{bmatrix}\) should result in:

• \( (2)(13) + (5)(-5) = 26 - 25 = 1 \),
• \( (2)(5) + (5)(-2) = 10 - 10 = 0 \),
• \( (-5)(13) + (-13)(-5) = -65 + 65 = 0 \),
• \( (-5)(5) + (-13)(-2) = -25 + 26 = 1 \).

The result is the identity matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), confirming the inverse is correct.

2x2 Matrix Inversion

A 2x2 matrix inversion is simpler than it might initially seem. This process involves a few key steps once you've ensured the matrix's determinant is non-zero:

• Calculate the matrix's determinant.
• Find the adjugate of the matrix.
• Multiply the adjugate by the reciprocal of the determinant.

So, let's say you've figured out that the determinant is \(-1\). Then, with the adjugate \(\begin{bmatrix} -13 & -5 \ 5 & 2 \end{bmatrix}\), and using the formula:\[ A^{-1} = \frac{1}{\text{det}(A)} \times \text{adjugate}(A) \]In this example, the inverse becomes:\[ \begin{bmatrix} 13 & 5 \ -5 & -2 \end{bmatrix} \]This simplicity of calculation makes 2x2 matrix inversion a useful technique in linear algebra.