Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{rrr} 3 & 0 & 2 \\ 0 & 1 & 0 \\ -4 & 0 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse is \[ \begin{bmatrix} \frac{1}{7} & 0 & \frac{-2}{7} \\ \frac{-4}{7} & \frac{1}{7} & 0 \\ 0 & 0 & \frac{3}{14} \end{bmatrix} \]

Step 1: Determine Matrix Dimensions

First, confirm that the matrix is square. The given matrix is a 3x3 matrix, which is square and eligible for inversion if its determinant is non-zero.

Step 2: Calculate the Determinant of the Matrix

To find the inverse, ensure the determinant is non-zero. For a 3x3 matrix \[A = \begin{bmatrix} 3 & 0 & 2 \ 0 & 1 & 0 \ -4 & 0 & 2 \end{bmatrix} \] the determinant \( \text{det}(A) \) is computed using the formula:\[ 3(1 \cdot 2 - 0 \cdot 0) - 0(0 \cdot 2 - (-4) \cdot 0) + 2((0 \cdot 0 - 1 \cdot (-4))) = 3(2) + 2(4) = 6 + 8 = 14 \] Since \( \text{det}(A) = 14 eq 0 \), the matrix is invertible.

Step 3: Find the Matrix of Minors

Compute the matrix of minors by finding the determinant of each 2x2 submatrix:\[ \begin{bmatrix} 1 \cdot 2 - 0 \cdot 0 & (0 \cdot 2 - (-4) \cdot 0) & 0 \cdot 0 - 1 \cdot (-4) \ 0 \cdot 2 - (-4) \cdot 2 & 3 \cdot 2 - 2 \cdot 2 & 3 \cdot 0 - 2 \cdot 0 \ 0 \cdot 0 - 0 \cdot 1 & 0 \cdot 2 - 0 \cdot 3 & 3 \cdot 1 - 0 \cdot 0 \end{bmatrix}\] simplifies to:\[ \begin{bmatrix} 2 & 0 & 4 \ 8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \]

Step 4: Formulate the Matrix of Cofactors

Apply the checkerboard pattern of signs (plus, minus, plus, etc.) to the matrix of minors:\[ \begin{bmatrix} +2 & 0 & -4 \ -8 & +2 & 0 \ 0 & 0 & +3 \end{bmatrix} \]

Step 5: Transpose the Cofactor Matrix

Transpose the cofactor matrix to obtain the adjugate matrix:\[ \begin{bmatrix} 2 & -8 & 0 \ 0 & 2 & 0 \ -4 & 0 & 3 \end{bmatrix} \] becomes\[ \begin{bmatrix} 2 & 0 & -4 \ -8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \]

Step 6: Calculate the Inverse Matrix

The inverse of matrix \(A\) is formed by dividing the adjugate matrix by the determinant (which is 14):\[ A^{-1} = \frac{1}{14} \begin{bmatrix} 2 & 0 & -4 \ -8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} = \begin{bmatrix} \frac{2}{14} & 0 & \frac{-4}{14} \ \frac{-8}{14} & \frac{2}{14} & 0 \ 0 & 0 & \frac{3}{14} \end{bmatrix} \]which simplifies to:\[ \begin{bmatrix} \frac{1}{7} & 0 & \frac{-2}{7} \ \frac{-4}{7} & \frac{1}{7} & 0 \ 0 & 0 & \frac{3}{14} \end{bmatrix} \]

Key Concepts

Determinant Calculation

The determinant of a matrix is a crucial value that determines whether a square matrix is invertible. For a 3x3 matrix, the formula for calculating the determinant involves taking the sum of the products of the elements of any row or column and their corresponding minor determinants, each multiplied by a sign depending on its position (positive or negative, known as the checkerboard pattern rule).

For example, for matrix \( A \) given as:

• \[ A = \begin{bmatrix} 3 & 0 & 2 \ 0 & 1 & 0 \ -4 & 0 & 2 \end{bmatrix} \]

the determinant \( \text{det}(A) \) is calculated by selecting any row or column, here, the first row for explanation purposes:

• Multiply the first element by the determinant of the 2x2 matrix formed from deleting the first row and first column.
• Continue this for each element in the row, applying the checkerboard sign pattern.
• Summing these calculations gives the determinant of the matrix.

By calculating, we find:

• \[ 3(1 \cdot 2 - 0 \cdot 0) + 0 + 2(0 \cdot 0 - 1 \cdot (-4)) = 6 + 8 = 14 \]

This non-zero value confirms that the matrix is invertible.

Matrix of Minors

The matrix of minors is obtained by computing the determinant of every possible 2x2 submatrix of the given 3x3 matrix. For each element in the original matrix, a corresponding minor matrix is created by eliminating the row and column of that element.

Calculating these minor determinants forms a new matrix known as the matrix of minors. For matrix \( A \), as shown below:

• \[ \begin{bmatrix} 1 \cdot 2 - 0 \cdot 0 & (0 \cdot 2 - (-4) \cdot 0) & 0 \cdot 0 - 1 \cdot (-4) \ 0 \cdot 2 - (-4) \cdot 2 & 3 \cdot 2 - 2 \cdot 2 & 3 \cdot 0 - 2 \cdot 0 \ 0 \cdot 0 - 0 \cdot 1 & 0 \cdot 2 - 0 \cdot 3 & 3 \cdot 1 - 0 \cdot 0 \end{bmatrix} \]

simplifies to:

• \[ \begin{bmatrix} 2 & 0 & 4 \ 8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \]

This matrix of minors is the foundational step toward determining the cofactor matrix.

Cofactor Matrix

Once the matrix of minors is established, the cofactor matrix is created by applying a checkerboard pattern of signs to it. This means that each entry in the matrix of minors is adjusted by a sign (positive or negative), depending on its position (even or odd sum of its row and column indices). This pattern starts with a `+` at the top-left position, alternating signs as you move across the matrix.

For instance, starting with our matrix of minors for matrix \( A \):

• Original Matrix of Minors: \[ \begin{bmatrix} 2 & 0 & 4 \ 8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \]
• Cofactor Matrix: \[ \begin{bmatrix} +2 & 0 & -4 \ -8 & +2 & 0 \ 0 & 0 & +3 \end{bmatrix} \]

This pattern is crucial for constructing the adjugate matrix in the inversion process.

Adjugate Matrix

The adjugate matrix (or adjoint) is the transpose of the cofactor matrix. The process of transposing involves swapping rows with columns, effectively flipping the matrix over its diagonal. This rearrangement is necessary as it enables the calculation of the matrix inverse when divided by the determinant.

• To transpose the cofactor matrix derived earlier for matrix \( A \): \[ \begin{bmatrix} 2 & 0 & -4 \ -8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \]
• Adjugate Matrix: \[ \begin{bmatrix} 2 & -8 & 0 \ 0 & 2 & 0 \ -4 & 0 & 3 \end{bmatrix} \]

This adjugate matrix is then divided by the determinant to obtain the inverse of the original matrix. In this example,

• \[ A^{-1} = \frac{1}{14} \cdot \begin{bmatrix} 2 & 0 & -4 \ -8 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} = \begin{bmatrix} \frac{1}{7} & 0 & \frac{-2}{7} \ \frac{-4}{7} & \frac{1}{7} & 0 \ 0 & 0 & \frac{3}{14} \end{bmatrix} \]

Concluding this process gives us the inverse matrix, completing the task of matrix inversion.