Problem 15 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The inverse matrix is \( \left[\begin{array}{rrr}-4 & -16 & 5 \\ -1 & 1 & 3 \\ 5 & 6 & -6\end{array}\right] \).

Step 1: Verify matrix is square

Check if the given matrix is a square matrix by ensuring the number of rows is equal to the number of columns. Here, the matrix is 3x3, which is square.

Step 2: Calculate the Determinant

To determine if the inverse exists, calculate the determinant of the matrix. The determinant is calculated as follows:\[\text{det}(A) = 2(1\cdot0 - (-1)\cdot4) - 4((-1)\cdot0 - (-1)\cdot1) + 1((-1)\cdot4 - 1\cdot1)\]\[= 2(4) - 4(1) + 1(-5) \]\[= 8 - 4 - 5 = -1\]The determinant is \(-1\), hence the inverse exists as it is non-zero.

Step 3: Find the Matrix of Minors

Compute the matrix of minors by finding the determinant of the minor matrices obtained by removing a specific row and column for each element. This results in:\[\begin{bmatrix}\det\begin{vmatrix}1 & -1 \ 4 & 0\end{vmatrix} & \det\begin{vmatrix}-1 & -1 \ 1 & 0\end{vmatrix} & \det\begin{vmatrix}-1 & 1 \ 1 & 4\end{vmatrix} \\det\begin{vmatrix}4 & 1 \ 4 & 0\end{vmatrix} & \det\begin{vmatrix}2 & 1 \ 1 & 0\end{vmatrix} & \det\begin{vmatrix}2 & 4 \ 1 & 4\end{vmatrix} \\det\begin{vmatrix}4 & 1 \ 1 & -1\end{vmatrix} & \det\begin{vmatrix}2 & 1 \ -1 & -1\end{vmatrix} & \det\begin{vmatrix}2 & 4 \ -1 & 1\end{vmatrix}\end{bmatrix} \]Calculating these gives:\[\begin{bmatrix}4 & -1 & -5 \-16 & -1 & 6 \-5 & 3 & 6\end{bmatrix}\]

Step 4: Find the Cofactor Matrix

Apply the correct sign pattern using \((-1)^{i+j}\) to the matrix of minors. The cofactor matrix is:\[\begin{bmatrix}4 & 1 & -5 \16 & -1 & -6 \-5 & -3 & 6\end{bmatrix}\]

Step 5: Transpose the Cofactor Matrix

Transpose the cofactor matrix to obtain the adjugate matrix. The transposed matrix is:\[\begin{bmatrix}4 & 16 & -5 \1 & -1 & -3 \-5 & -6 & 6\end{bmatrix}\]

Step 6: Multiply by Reciprocal of Determinant

Multiply the adjugate matrix by the reciprocal of the original determinant, \(-1\). This will give the inverse of the original matrix:\[\begin{bmatrix}-4 & -16 & 5 \-1 & 1 & 3 \5 & 6 & -6\end{bmatrix}\]

Step 7: Final Step: Result

Thus, the inverse matrix is:\[\begin{bmatrix}-4 & -16 & 5 \-1 & 1 & 3 \5 & 6 & -6\end{bmatrix}\]

Key Concepts

Determinant of a Matrix

The determinant is a special value that can be computed from a square matrix. The determinant provides essential information about the matrix, including whether or not the matrix has an inverse. If the determinant is zero, the matrix does not have an inverse, meaning it is singular. Conversely, if the determinant is not zero, the matrix can be inverted.

In the case of a 3x3 matrix, calculating the determinant involves a specific formula:

• Start with the first row and use each entry as a multiplier to the determinant of a 2x2 matrix.
• The 2x2 matrices are formed by removing the row and column of the current element being used as a multiplier.
• The formula also involves alternating signs, starting with a positive sign for the first element.

This might sound complicated, but it is essentially breaking the matrix down step-by-step into smaller, more manageable pieces. Calculating the determinant correctly is crucial because it determines the path forward when finding an inverse.

Matrix of Minors

After calculating the determinant and ensuring it's non-zero, the next step in finding a matrix inverse is to compute the matrix of minors. The matrix of minors is created by substituting each element of the original matrix with the determinant of a smaller 2x2 matrix.

Here's how you form the matrix of minors:

• For each element in the 3x3 matrix, temporarily remove the row and the column containing that element.
• Calculate the determinant of the remaining 2x2 sub-matrix.
• Replace the original element with this determinant.

Creating the matrix of minors forms the groundwork for the next step in finding a matrix inverse, which is the cofactor matrix.

Cofactor Matrix

Once you have the matrix of minors, you'll need to convert it into the cofactor matrix. The cofactor matrix is similar to the matrix of minors but includes a crucial step involving signs. This sign adjustment comes from the checkerboard pattern of plus and minus signs, which depends on the position of each element.

To create the cofactor matrix:

• Use the corresponding minor value from the matrix of minors.
• Apply a sign determined by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices of the element.

The cofactor matrix is essential because the transpose of this matrix, called the adjugate matrix, is what ultimately gets multiplied by the reciprocal of the determinant to find the inverse of the original matrix.

Transpose of a Matrix

The transpose of a matrix involves flipping the matrix over its diagonal. This operation switches the row and column indices of each element. The primary purpose of transposing the cofactor matrix in the process of finding an inverse is to form the adjugate matrix.

Steps to transpose a matrix:

• Consider each element \(a_{ij}\) in the matrix.
• Swap the row and column indices, so it becomes \(a_{ji}\).

For example, in a 3x3 matrix, the element at position (1,2) becomes the element at position (2,1) after transposition. The adjugate matrix, which is simply the transpose of the cofactor matrix, is then used alongside the determinant to solve for the inverse. This transposed version essentially prepares the matrix to be scaled by the determinant's reciprocal, yielding the final inverse matrix.