Problem 8 - Precalculus: Mathematical for Calculus

Question

Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll}3 & 4 \\ 7 & 9\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} -9 & 4 \\ 7 & -3 \end{bmatrix} \).

Step 1: Calculate the Determinant

To find the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), we need to first calculate the determinant. For the matrix \( \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \), the determinant \( \Delta \) is given by the formula: \[ \Delta = ad - bc = (3)(9) - (4)(7) \]Calculating this, we have:\[ \Delta = 27 - 28 = -1 \]

Step 2: Check Invertibility

A matrix is invertible if its determinant is not zero. Since we found that the determinant of the matrix is \( -1 \), which is not zero, the matrix is invertible.

Step 3: Use the Adjugate Matrix Formula

Once we know the matrix is invertible, we use the inverse formula for a 2x2 matrix: \[ A^{-1} = \frac{1}{\Delta} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]Where \( a = 3 \), \( b = 4 \), \( c = 7 \), \( d = 9 \), and \( \Delta = -1 \). The adjugate (also called adjoint) of the matrix is:\[ \begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix} \]

Step 4: Calculate the Inverse Matrix

To find the inverse of the matrix, divide each term of the adjugate matrix by the determinant \( \Delta \):\[ A^{-1} = \frac{1}{-1} \begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix} = \begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix} \]Thus, the inverse of the given matrix is:\[ \begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix} \]

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements and provides crucial insights into the properties of the matrix. For a 2x2 matrix, such as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \Delta \) is found using the formula:

• \( \Delta = ad - bc \)

In this formula, \( a, b, c, \) and \( d \) are the elements of the matrix. The determinant calculation is an essential first step when finding the inverse of a matrix. If the determinant is zero, the matrix is termed singular and has no inverse. In our exercise\(
\)for the matrix \( \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \), we calculated the determinant as:

• \( \Delta = (3)(9) - (4)(7) = 27 - 28 = -1 \)

This result indicates that the determinant is not zero, hence the matrix might be invertible.

Invertible Matrix

A matrix is said to be invertible if it has an inverse, which mathematically means that for a matrix \( A \), there exists another matrix \( A^{-1} \) such that:

• \( A \cdot A^{-1} = I \)

Here, \( I \) is the identity matrix. The inverse of a matrix is significant in solving systems of equations, among other applications. However, not all matrices can be inverted. The key condition we look for is checking whether the determinant is non-zero. If the determinant of a matrix is zero, the matrix is non-invertible (or singular).
In our context, since the determinant of the matrix \( \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \) is \( -1 \), which is not zero, the matrix is invertible. This opens the way for further steps in calculating its inverse.

2x2 Matrix Inverse

The inverse of a 2x2 matrix can be found using a specific formula that relies on its determinant and its adjugate matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula to find the inverse \( A^{-1} \) is given by:

• \( A^{-1} = \frac{1}{\Delta} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \)

The formula involves two main components: the determinant \( \Delta \) and the adjugate matrix \( \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). This formula is handy because it provides a consistent way to calculate the inverse by simply rearranging and scaling the original matrix elements.
Using our determinant \( \Delta = -1 \), and the elements from matrix \( \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \), we find:

• The inverse is \( \begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix} \)

This matrix \( \begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix} \) is the exact opposite in terms of multiplication to the original matrix, confirming it is indeed its inverse.

Adjugate Matrix

The adjugate matrix, sometimes referred to as the adjoint, plays a key role in computing the inverse of a matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the adjugate is created by swapping the positions of the diagonal elements \( a \) and \( d \), and changing the signs of the off-diagonal elements \( b \) and \( c \):

• The adjugate matrix is \( \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).

This simple rearrangement and sign change are crucial because the inverse formula uses the adjugate matrix as a central component. For the example matrix \( \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \), we find the adjugate matrix to be \( \begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix} \), which is then scaled by the determinant to find the inverse. Understanding the role of the adjugate matrix helps in quickly assessing properties and feasibility for matrix inversion.