Problem 10 - Precalculus Mathematics 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 of the matrix is \(\begin{bmatrix} -9 & 4 \\ 7 & -3 \end{bmatrix}\).

Step 1: Check if the Matrix is Invertible

First, we need to check if the matrix is invertible by calculating its determinant. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is given by \(ad - bc\). For the matrix \(\begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix}\), the determinant is calculated as follows: \(3 \cdot 9 - 4 \cdot 7 = 27 - 28 = -1\). Since the determinant is not zero, the matrix is invertible.

Step 2: Find the Adjugate of the Matrix

The next step is to find the adjugate (adjoint) of the matrix. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the adjugate is \(\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\). Thus, the adjugate of \(\begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix}\) is \(\begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix}\).

Step 3: Calculate the Inverse of the Matrix

To find the inverse of a matrix, we divide the adjugate by the determinant. The formula is: \[ A^{-1} = \frac{1}{\text{det}} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] Therefore, the inverse of \(\begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix}\) is \[ \frac{1}{-1} \begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix} = \begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix} \]. Thus, the inverse is \(\begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix}\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix. It is vital in determining whether a matrix is invertible or not. For a 2x2 matrix, \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant is calculated as \( ad - bc \).

• If the determinant is zero, the matrix is not invertible and doesn't have an inverse.
• If the determinant is non-zero, the matrix is invertible, and we can find its inverse.

In the given exercise, \[ \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \] has the determinant \( 3 \cdot 9 - 4 \cdot 7 = 27 - 28 = -1 \). Since the determinant is \(-1\), which is not zero, the matrix is indeed invertible. This allows us to proceed to find its inverse.

Adjugate of a Matrix

The adjugate of a matrix, sometimes called the adjoint, is crucial when calculating the inverse of a matrix. For a 2x2 matrix, \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the adjugate is formed by swapping the positions of \( a \) and \( d \), and changing the signs of \( b \) and \( c \). This results in the matrix: \[ \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]. Applying this to our matrix \[ \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \], we swap \( 3 \) and \( 9 \), and change the signs for \( 4 \) and \( 7 \), giving us the adjugate matrix: \[ \begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix} \]. The adjugate is an essential step before computing the inverse, as we'll divide this matrix by the determinant in the next step.

Invertible Matrix

An invertible matrix, also known as a nonsingular or non-degenerate matrix, is one that has an inverse. For a matrix to be invertible, its determinant must be non-zero. The inverse of a matrix is the matrix that, when multiplied with the original matrix, yields the identity matrix. The identity matrix for a 2x2 matrix looks like \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]. To find the inverse of an invertible matrix \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], we use the formula: \[ A^{-1} = \frac{1}{\text{det}} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]. For our matrix \[ \begin{bmatrix} 3 & 4 \ 7 & 9 \end{bmatrix} \], with a determinant of \(-1\), its inverse is calculated as:\[ \frac{1}{-1} \begin{bmatrix} 9 & -4 \ -7 & 3 \end{bmatrix} = \begin{bmatrix} -9 & 4 \ 7 & -3 \end{bmatrix} \]. The resulting inverse matrix can be verified by multiplying it with the original matrix to see if the identity matrix is obtained, ensuring correctness of our inversion.