Problem 22 - College Algebra

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$\left[\begin{array}{rrr}1 & 2 & -1 \\ -3 & 4 & 1 \\ -2 & -4 & -5\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the matrix is \( \frac{1}{-54} \) times the adjugate matrix.

Step 1: Determine if the inverse exists

We need to calculate the determinant of the matrix to check if an inverse exists. The determinant is given by: \[ \text{det}(A) = 1(4(-5) - 1(-4)) - 2(-3(-5) - 1(-2)) - 1(-3(-4) - 4(-2))\]Simplifying, we have:\[= 1(-20 + 4) - 2(15 + 2) - 1(12 - 8)\]\[= 1(-16) - 2(17) - 1(4)\]\[= -16 - 34 - 4 = -54\]Since the determinant is not zero, the inverse exists.

Step 2: Find the Adjugate Matrix

The adjugate of a matrix is the transpose of the cofactor matrix. We calculate cofactors for each entry of the matrix and arrange them into a matrix:\(C = \begin{bmatrix}\begin{vmatrix} 4 & 1 \ -4 & -5 \end{vmatrix} & - \begin{vmatrix} -3 & 1 \ -2 & -5 \end{vmatrix} & \begin{vmatrix} -3 & 4 \ -2 & -4 \end{vmatrix} \- \begin{vmatrix} 2 & -1 \ -4 & -5 \end{vmatrix} & \begin{vmatrix} 1 & -1 \ -2 & -5 \end{vmatrix} & - \begin{vmatrix} 1 & 2 \ -2 & -4 \end{vmatrix} \\begin{vmatrix} 2 & -1 \ 4 & 1 \end{vmatrix} & - \begin{vmatrix} 1 & -1 \ -3 & 1 \end{vmatrix} & \begin{vmatrix} 1 & 2 \ -3 & 4 \end{vmatrix}\end{bmatrix}\)Calculating, we find:\(C = \begin{bmatrix}4 & 23 & -20 \6 & -7 & -8 \5 & -2 & 10\end{bmatrix}\)The adjugate matrix is the transpose of this matrix.

Step 3: Calculate the Inverse Matrix

The inverse of the matrix, if it exists, is given by:\[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\]We already found \(\text{det}(A) = -54\) and the adjugate matrix as:\[\text{adj}(A) = \begin{bmatrix} 4 & 6 & 5 \ 23 & -7 & -2 \ -20 & -8 & 10 \end{bmatrix}\]Thus,\[A^{-1} = \frac{1}{-54} \cdot \begin{bmatrix} 4 & 6 & 5 \ 23 & -7 & -2 \ -20 & -8 & 10 \end{bmatrix}\]Simplifying, this matrix gives the final inverse matrix.

Key Concepts

Determinant Calculation

Before diving into finding the inverse of a matrix, we must first determine if the inverse exists by calculating the determinant. The determinant is a special number that you can compute from a square matrix. It is essential because if the determinant of the matrix is zero, the inverse does not exist.
For our matrix, we begin by using the formula for a 3x3 matrix determinant, which involves a mixture of multiplication and addition/subtraction of the elements. Here's the formula:

• For a 3x3 matrix: \[det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]
• You replace the letters a, b, c, etc., with the corresponding elements of your matrix.

In the exercise, after performing the necessary arithmetic operations step-by-step, we find that the determinant of our matrix is -54. Since the determinant is not zero, the matrix has an inverse.

Cofactor Matrix

To find the inverse of a matrix, we first need to construct a cofactor matrix. This involves calculating the cofactor for each element. Each cofactor is essentially the determinant of a minor matrix - which you get by removing the row and column of the element from the original matrix - multiplied by \( (-1)^{i+j} \), where \(i\) and \(j\) are the row and column numbers.

• Identify the minor for each element in the matrix.
• Calculate its determinant.
• Multiply by \( (-1)^{i+j} \) to get the cofactor.

In our example, every cofactor is calculated this way, resulting in a cofactor matrix of:\[C = \begin{bmatrix}4 & 23 & -20 \6 & -7 & -8 \5 & -2 & 10 \end{bmatrix}\]This matrix becomes the basis for creating the adjugate matrix.

Adjugate Matrix

The adjugate of a matrix is formed by taking the transpose of the cofactor matrix. Transposing involves flipping the matrix over its diagonal, essentially switching its rows and columns. This step is crucial because the adjugate matrix is directly used in the calculation of the inverse.
To transpose a 3x3 matrix, the process is straightforward and goes as follows:

• Swap the element at position (1,2) with the element at position (2,1).
• Swap the element at position (1,3) with the element at position (3,1).
• Swap the element at position (2,3) with the element at position (3,2).

For our cofactor matrix, the transposed version is:\[adj(A) = \begin{bmatrix}4 & 6 & 5 \23 & -7 & -2 \-20 & -8 & 10\end{bmatrix}\]This adjugate matrix is what you'll multiply by the reciprocal of the determinant to get the inverse.

Inverse Matrix Formula

The inverse of a matrix is the matrix that, when multiplied with the original matrix, results in the identity matrix. To find the inverse, two key components are needed: the determinant and the adjugate matrix.

• The determinant helps decide the existence and scaling effect of the matrix.
• The adjugate matrix sets the structure and orientation of the inverse.

The inverse matrix formula is:\[A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\]For our matrix, with \det(A)=-54\ and the calculated adjugate matrix, the inverse is:\[A^{-1} = \frac{1}{-54} \cdot \begin{bmatrix}4 & 6 & 5 \23 & -7 & -2 \-20 & -8 & 10\end{bmatrix}\]By multiplying each element of the adjugate matrix by \(-\frac{1}{54}\), you find the inverse matrix. This final matrix can be used to solve systems of equations and other mathematical calculations requiring matrix inversion.