Problem 20 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rrr} -2 & 1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix}-1 & 0 & 1 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{bmatrix} \).

Step 1: Check the Determinant

Calculate the determinant of matrix \( A \) to determine if the inverse exists. For a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant \( \det(A) \) is given by:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Apply this formula using: \( a = -2 \), \( b = 1 \), \( c = 0 \), \( d = 1 \), \( e = 0 \), \( f = 1 \), \( g = -1 \), \( h = 1 \), \( i = 0 \).\[\det(A) = -2(0 \cdot 0 - 1 \cdot 1) - 1(1 \cdot 0 - (-1) \cdot 1) + 0(1 \cdot 1 - 0 \cdot (-1))\]\[= -2(0 - 1) - 1(0 + 1) + 0\]\[= 2 - 1\]\[= 1\]Since the determinant is non-zero (\( \det(A) = 1 \)), matrix \( A \) has an inverse.

Step 2: Find the Adjugate Matrix

To find \( A^{-1} \), we first need the cofactors of each element of \( A \), and then the adjugate (transpose of the cofactor matrix).Cofactors:- \( C_{11} = \det\begin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix} = 0 - 1 = -1\)- \( C_{12} = -\det\begin{bmatrix}1 & 1 \ -1 & 0\end{bmatrix} = -(0 + 1) = -1\)- \( C_{13} = \det\begin{bmatrix}1 & 0 \ -1 & 1\end{bmatrix} = 1 - 0 = 1\)- \( C_{21} = -\det\begin{bmatrix}1 & 0 \ -1 & 0\end{bmatrix} = -0 = 0\)- \( C_{22} = \det\begin{bmatrix}-2 & 0 \ -1 & 0\end{bmatrix} = 0 - 0 = 0\)- \( C_{23} = -\det\begin{bmatrix}-2 & 1 \ -1 & 1\end{bmatrix} = -(1 + 2) = -3\)- \( C_{31} = \det\begin{bmatrix}1 & 0 \ 1 & 1\end{bmatrix} = 1 - 0 = 1\)- \( C_{32} = -\det\begin{bmatrix}-2 & 0 \ 1 & 1\end{bmatrix} = -((-2) - 0) = 2\)- \( C_{33} = \det\begin{bmatrix}-2 & 1 \ 1 & 0\end{bmatrix} = 0 - 1 = -1\)Cofactor Matrix:\[\begin{bmatrix}-1 & -1 & 1 \0 & 0 & -3 \1 & 2 & -1\end{bmatrix}\]Adjugate Matrix (transpose of the cofactor matrix):\[\begin{bmatrix}-1 & 0 & 1 \-1 & 0 & 2 \1 & -3 & -1\end{bmatrix}\]

Step 3: Calculate the Inverse Matrix

Using the formula \( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \), where \( \text{adj}(A) \) is the adjugate matrix and \( \det(A) = 1 \).\[A^{-1} = 1 \cdot \begin{bmatrix}-1 & 0 & 1 \-1 & 0 & 2 \1 & -3 & -1\end{bmatrix} = \begin{bmatrix}-1 & 0 & 1 \-1 & 0 & 2 \1 & -3 & -1\end{bmatrix}\]Thus, the inverse of matrix \( A \) is:\[A^{-1} = \begin{bmatrix}-1 & 0 & 1 \-1 & 0 & 2 \1 & -3 & -1\end{bmatrix}\]

Key Concepts

Determinant Calculation

The determinant of a matrix is an important value in linear algebra. It helps in determining whether a matrix has an inverse. The determinant can be thought of as a scalar value that describes certain properties of the matrix, like volume scaling or system solvability. In the case of a 3x3 matrix, the determinant is found using a specific formula. For a matrix \( A \) with elements:

• \[ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]

The determinant \( \det(A) \) is calculated using:

• \[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

This equation involves multiplying and subtracting elements of the matrix in a way that captures the essence of the linear transformations described by the matrix. If the determinant is zero, the matrix does not have an inverse. If it is non-zero, as in our example where \( \det(A) = 1 \), the matrix does have an inverse.

Cofactor Matrix

Once we have confirmed that the matrix has an inverse, the next step is to calculate the cofactors. Each element of the original matrix is replaced by its cofactor, which involves considering the minor matrix.

A minor is the determinant of a smaller matrix that remains when removing one row and one column from the original. The cofactor is calculated by multiplying the minor by \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column indices, respectively.

The formation of a cofactor matrix is crucial in finding the adjugate. In our example:

• The cofactor of the top-left element \( C_{11} \) involves excluding the first row and column and calculating the determinant of the resulting 2x2 matrix, producing \(-1\).
• Similarly, other cofactors are determined for each matrix element, forming the cofactors' matrix:\[\begin{bmatrix}-1 & -1 & 1 \0 & 0 & -3 \1 & 2 & -1\end{bmatrix}\]

Each cofactor element reflects a weighted measure of specific interactions within the matrix.

Adjugate Matrix

The adjugate matrix, also known as the adjoint, is derived from the cofactor matrix. To find the adjugate matrix, we transpose the cofactor matrix.

Transposing involves flipping the matrix over its diagonal, which switches the row and column indices of the elements. This creates a new matrix where each element is the cofactor of the respective position in the original matrix, but positioned according to the transpose operation.

In our example, the cofactor matrix:

• \[\begin{bmatrix}-1 & -1 & 1 \0 & 0 & -3 \1 & 2 & -1\end{bmatrix}\] is transposed to:\[\begin{bmatrix}-1 & 0 & 1 \-1 & 0 & 2 \1 & -3 & -1\end{bmatrix}\]

This adjugate matrix is used to compute the final inverse by multiplying it with the reciprocal of the determinant.

Inverse Matrix

Finding an inverse matrix is important for solving linear equations and other mathematical problems. An inverse exists only for non-singular matrices, those with a non-zero determinant, as seen earlier.

The inverse matrix \( A^{-1} \) is calculated using:

• \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \]

This involves multiplying each element of the adjugate matrix by \( \frac{1}{\det(A)} \), the reciprocal of the determinant.

Because our matrix has a determinant of 1, its inverse is:

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

The inverse, when multiplied by the original matrix, results in the identity matrix, confirming its correctness.