Problem 24 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rrr}3 & -2 & 0 \\\5 & 1 & 1 \\\2 & -2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of the given matrix is \(\begin{bmatrix} 1 & 0 & -1 \\ 1 & 0 & -1.5 \\ -6 & 1 & 6.5 \end{bmatrix}.\)

Step 1: Check if the Inverse Exists (Determinant)

First, you need to determine if the matrix has an inverse by calculating its determinant. The given matrix is\[A = \begin{bmatrix} 3 & -2 & 0 \ 5 & 1 & 1 \ 2 & -2 & 0 \end{bmatrix}\]For a 3x3 matrix, the determinant \( \text{det}(A) \) is calculated using:\[\text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]Substitute the values:\[\text{det}(A) = 3(1 \cdot 0 - 1 \cdot -2) - (-2)(5 \cdot 0 - 1 \cdot 2) + 0(5 \cdot -2 - 1 \cdot 2)\]\[= 3(2) - (-2)(-2) + 0\]\[= 6 - 4 = 2\]Since \( \text{det}(A) eq 0 \), the inverse exists.

Step 2: Find the Matrix of Minors

The first step in finding the inverse is computing the matrix of minors. Each element of the matrix of minors is the determinant of the 2x2 submatrix obtained by removing the row and column of that element from the original matrix. Calculate them for each element:\[\text{Minor of A}_{11} = \begin{vmatrix} 1 & 1 \ -2 & 0 \end{vmatrix} = (1)(0) - (1)(-2) = 2\]\[\text{Minor of A}_{12} = \begin{vmatrix} 5 & 1 \ 2 & 0 \end{vmatrix} = (5)(0) - (1)(2) = -2\]\[\text{Minor of A}_{13} = \begin{vmatrix} 5 & 1 \ 2 & -2 \end{vmatrix} = (5)(-2) - (1)(2) = -12\]\[\text{Minor of A}_{21} = \begin{vmatrix} -2 & 0 \ -2 & 0 \end{vmatrix} = (-2)(0) - (0)(-2) = 0\]\[\text{Minor of A}_{22} = \begin{vmatrix} 3 & 0 \ 2 & 0 \end{vmatrix} = (3)(0) - (0)(2) = 0\]\[\text{Minor of A}_{23} = \begin{vmatrix} 3 & -2 \ 2 & -2 \end{vmatrix} = (3)(-2) - (-2)(2) = -2\]\[\text{Minor of A}_{31} = \begin{vmatrix} -2 & 0 \ 1 & 1 \end{vmatrix} = (-2)(1) - (0)(1) = -2\]\[\text{Minor of A}_{32} = \begin{vmatrix} 3 & 0 \ 5 & 1 \end{vmatrix} = (3)(1) - (0)(5) = 3\]\[\text{Minor of A}_{33} = \begin{vmatrix} 3 & -2 \ 5 & 1 \end{vmatrix} = (3)(1) - (-2)(5) = 3 + 10 = 13\]

Step 3: Form the Cofactor Matrix

The cofactor of each element is found by applying a sign to each minor. The sign is determined by \((-1)^{i+j}\) for an element in the \(i\)-th row and \(j\)-th column.\[\text{Cofactor Matrix} = \begin{bmatrix} 2 & 2 & -12 \ 0 & 0 & 2 \ -2 & -3 & 13 \end{bmatrix}\]

Step 4: Transpose the Cofactor Matrix

Transpose the cofactor matrix to find the adjugate matrix. This involves swapping rows with columns:\[\text{Adjugate of A} = \begin{bmatrix} 2 & 0 & -2 \ 2 & 0 & -3 \ -12 & 2 & 13 \end{bmatrix}\]

Step 5: Compute the Inverse Using the Determinant

The inverse of the matrix is calculated by dividing the adjugate matrix by the determinant of the original matrix:\[A^{-1} = \frac{1}{2} \begin{bmatrix} 2 & 0 & -2 \ 2 & 0 & -3 \ -12 & 2 & 13 \end{bmatrix} = \begin{bmatrix} 1 & 0 & -1 \ 1 & 0 & -1.5 \ -6 & 1 & 6.5 \end{bmatrix}\]

Key Concepts

Determinant

Finding the determinant is a crucial step in determining if a matrix has an inverse. The determinant gives us a measure of the matrix's singularity. For a 3x3 matrix, the determinant is calculated by selecting a row or column and using the formula:

• Det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This method involves breaking down the matrix into smaller 2x2 sections and calculating their determinants, then combining them with specific arithmetic operations related to the elements of the selected row or column. If the determinant equals zero, the matrix does not have an inverse. In our case, however, it equals 2, which is not zero, thus confirming the inverse exists.

Matrix of Minors

The matrix of minors is the next step in finding a matrix's inverse. Each element of this matrix is derived by removing the corresponding row and column from the original matrix and calculating the determinant of the remaining 2x2 matrix.

• This determinant is known as a minor.

For the given example, each element's minor is calculated, resulting in a new matrix. Understanding the matrix of minors is important as it helps to build towards the cofactor matrix, which is an intermediate step before finding the adjugate matrix. Practicing the calculation of minors improves your familiarity with matrix manipulation.

Cofactor Matrix

Once we obtain the matrix of minors, we move on to forming the cofactor matrix. This involves applying a specific sign to each minor.

• The sign is determined by the formula \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number.

Using this pattern, you assign positive or negative signs to each minor, transforming the matrix of minors into the cofactor matrix. This step prepares the matrix for transposition into the adjugate matrix. Creating a cofactor matrix helps in organizing and simplifying the determinants involved in finding the matrix inverse.

Adjugate Matrix

The adjugate matrix is found by transposing the cofactor matrix. Transposing a matrix involves switching its rows with its columns. In essence, you're rotating the matrix so that wherever you had a certain row, you now have a corresponding column, and vice versa.
This is crucial because the adjugate matrix is directly used to calculate the inverse of the original matrix.

• The formula for an inverse matrix is A\(^{-1}\) = \(\frac{1}{det(A)}\) * (adjugate of A).

Here, multiplying your adjugate matrix by the reciprocal of the original matrix's determinant gives you the inverse. Understanding and applying the concept of the adjugate matrix is a vital step in mastering matrix operations and solving linear equation systems efficiently.