Problem 30 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 5 & -3 & 2 \\ -5 & 3 & -2 \\ 1 & 0 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix A exists and is \( \begin{bmatrix} -\frac{1}{4} & -\frac{1}{4} & \frac{1}{12} \\ -\frac{5}{12} & -\frac{1}{4} & 0 \\ \frac{1}{4} & \frac{1}{4} & 0 \end{bmatrix} \).

Step 1: Calculate the Determinant of Matrix A

To determine if the inverse of the matrix exists, we first calculate the determinant of \( A \). The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \) is given by \( \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \). For matrix \( A = \begin{bmatrix} 5 & -3 & 2 \ -5 & 3 & -2 \ 1 & 0 & 1 \end{bmatrix} \), calculate: \[ \det(A) = 5(3 \cdot 1 - (-2) \cdot 0) - (-3)(-5 \cdot 1 - (-2) \cdot 1) + 2(-5 \cdot 0 - 3 \cdot 1) \] This simplifies to: \[ \det(A) = 5(3) - 3(5 + 2) + 2(0 - 3) \] \[ = 15 - 21 - 6 = -12 \] Since the determinant is not zero, \( A^{-1} \) exists.

Step 2: Form the Matrix of Minors

The next step is to calculate the matrix of minors for \( A \). The minor of each element in a matrix is the determinant of the submatrix formed by deleting the row and column of that element. The matrix of minors for \( A \) is: \[ \begin{bmatrix} \det\begin{bmatrix} 3 & -2 \ 0 & 1 \end{bmatrix} & \det\begin{bmatrix} -5 & -2 \ 1 & 1 \end{bmatrix} & \det\begin{bmatrix} -5 & 3 \ 1 & 0 \end{bmatrix} \ \det\begin{bmatrix} -3 & 2 \ 0 & 1 \end{bmatrix} & \det\begin{bmatrix} 5 & 2 \ 1 & 1 \end{bmatrix} & \det\begin{bmatrix} 5 & -3 \ 1 & 0 \end{bmatrix} \ \det\begin{bmatrix} -3 & 2 \ 3 & -2 \end{bmatrix} & \det\begin{bmatrix} 5 & 2 \ -5 & -2 \end{bmatrix} & \det\begin{bmatrix} 5 & -3 \ -5 & 3 \end{bmatrix} \end{bmatrix} \] Calculating each minor: \[ \begin{bmatrix} 3 & -5 & -3 \ -3 & 3 & 3 \ -1 & 0 & 0 \end{bmatrix} \]

Step 3: Form the Cofactor Matrix

To form the cofactor matrix, apply the checkerboard pattern of signs to the matrix of minors: \( \begin{bmatrix} + & - & + \ - & + & - \ + & - & + \end{bmatrix} \). Thus, the cofactor matrix is: \[ \begin{bmatrix} 3 & 5 & -3 \ 3 & 3 & -3 \ -1 & 0 & 0 \end{bmatrix} \]

Step 4: Transpose the Cofactor Matrix

The next step is to transpose the cofactor matrix to find the adjugate matrix: Transpose the cofactor matrix: \[ \begin{bmatrix} 3 & 3 & -1 \ 5 & 3 & 0 \ -3 & -3 & 0 \end{bmatrix} \]

Step 5: Calculate the Inverse of Matrix A

The inverse of a matrix \( A \) is given by \( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \). We've already found \( \det(A) = -12 \) and the adjugate matrix. So, Multiply each element of the adjugate matrix by \( \frac{1}{-12} \): \[ A^{-1} = \frac{1}{-12} \begin{bmatrix} 3 & 3 & -1 \ 5 & 3 & 0 \ -3 & -3 & 0 \end{bmatrix} = \begin{bmatrix} -\frac{1}{4} & -\frac{1}{4} & \frac{1}{12} \ -\frac{5}{12} & -\frac{1}{4} & 0 \ \frac{1}{4} & \frac{1}{4} & 0 \end{bmatrix} \] This is the inverse of matrix \( A \).

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can tell us many things about the matrix, one being if an inverse exists. For a 3x3 matrix, the determinant is calculated using a particular formula. It involves multiplying and subtracting products of matrix elements. The calculation for our given matrix \[A=\left[\begin{array}{rrr} 5 & -3 & 2 \ -5 & 3 & -2 \ 1 & 0 & 1 \end{array}\right]\]goes like this:

• First, multiply the top-left element by the determinant of the bottom-right 2x2 matrix formed by removing the row and column of our selected element.
• Do similar calculations for the two other elements of the top row, keeping the signs in mind (positive, negative, positive).

For matrix \(A\), the determinant \[\det(A) = 5(3 \times 1 - (-2) \times 0) - (-3)(-5 \times 1 - (-2) \times 1) + 2(-5 \times 0 - 3 \times 1).\]This results in a determinant of \(-12\). Since it's not zero, \(A^{-1}\) exists, making further calculations possible!

Matrix of Minors

A matrix of minors is created by finding the minor for each element in the matrix. The minor of an element is the determinant of the smaller matrix that remains after removing the row and column of that element. For matrix \(A\), we systematically find the minors:

• For the first element, 5, remove the first row and column and calculate the determinant of the resulting 2x2 matrix.
• Repeat this for every element in the matrix, ensuring that each minor is calculated from its respective remaining elements.

This process results in a new matrix filled with these minors. Our matrix of minors for \(A\) is:\[\begin{bmatrix}3 & -5 & -3 \-3 & 3 & 3 \-1 & 0 & 0\end{bmatrix}\]Understanding minors is crucial as they form the basis for creating cofactors, which are pivotal for calculating the inverse.

Cofactor Matrix

Once you have the matrix of minors, transforming it into a cofactor matrix involves applying a pattern of signs, often called a checkerboard pattern. These signs alternate between positive and negative, starting with a positive sign at the top-left corner of the matrix.

• The cofactor for each element is simply its minor multiplied by the corresponding sign from the checkerboard pattern.

For our matrix \(A\), the matrix of minors was as follows:\[\begin{bmatrix}3 & -5 & -3 \-3 & 3 & 3 \-1 & 0 & 0\end{bmatrix}\]Applying the checkerboard pattern results in the cofactor matrix:\[\begin{bmatrix}3 & 5 & -3 \3 & 3 & -3 \-1 & 0 & 0\end{bmatrix}\]This matrix helps in finding the adjugate, which is a key step toward determining the inverse of the original matrix.

Adjugate Matrix

The adjugate matrix is derived by transposing the cofactor matrix. Transposition involves switching the matrix's rows and columns. For our matrix \(A\), let’s take the obtained cofactor matrix:\[\begin{bmatrix}3 & 5 & -3 \3 & 3 & -3 \-1 & 0 & 0\end{bmatrix}\]To get the adjugate,

• Swap the elements over the main diagonal, moving each element to its transposed position.

This action produces the adjugate matrix:\[\begin{bmatrix}3 & 3 & -1 \5 & 3 & 0 \-3 & -3 & 0\end{bmatrix}\]Using this adjugate and the previously calculated determinant, the inverse matrix can finally be determined. Remember, every step is interconnected, and understanding how they build upon one another is vital to mastering matrix inverses.