Problem 27 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{ccc} -0.4 & 0.1 & 0.2 \\ 0 & 0.6 & 0.8 \\ 0.3 & 0 & -0.2 \end{array}\right]$$

Step-by-Step Solution

Answer

Matrix \(A\) is invertible with an inverse given by \(A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)\).

Step 1: Check if the matrix is square

To find the inverse of a matrix \(A\), we first need to check if the matrix is square. Matrix \(A\) is 3x3, so it is square.

Step 2: Calculate the determinant of matrix A

For a 3x3 matrix \(A\) with elements \(a_{ij}\), the determinant is calculated using the formula: \(\text{det}(A) = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\). For matrix \(A\), calculate: \[\text{det}(A) = (-0.4)(0.6 \times -0.2 - 0.8 \times 0) - (0.1)(0 \times -0.2 - 0.8 \times 0.3) + (0.2)(0 \times 0 - 0.6 \times 0.3)\ = -0.048 - 0 + 0.036 = -0.012\] The determinant of \(A\) is \(-0.012\).

Step 3: Verify if the determinant is non-zero

Since the determinant of matrix \(A\) is \(-0.012\), which is non-zero, it indicates that \(A\) has an inverse.

Step 4: Find the adjugate of matrix A

The adjugate of a 3x3 matrix is obtained by taking the transpose of the cofactor matrix. Calculate the cofactor for each element, and then take the transpose to get the adjugate matrix: 1. Compute cofactors for each entry. 2. Form the matrix of cofactors. 3. Transpose the matrix of cofactors to get the adjugate.

Step 5: Compute the inverse of matrix A using the formula

The inverse of matrix \(A\) is given by the formula \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \). Use the determinant \(-0.012\) and the adjugate matrix to find the inverse.

Key Concepts

Determinant Calculation

To find the inverse of a matrix, we first calculate its determinant. The determinant is a special number that gives us important information about the matrix. For a 3x3 matrix like \(A\), the determinant \(\text{det}(A)\) is calculated using a specific formula:

• Formula: \(\text{det}(A) = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\)

In our example, matrix \(A\) is:\[A=\begin{bmatrix} -0.4 & 0.1 & 0.2 \0 & 0.6 & 0.8 \0.3 & 0 & -0.2 \end{bmatrix}\]Plug these values into the formula to find:

• \(\text{det}(A) = (-0.4)(0.6 \times -0.2 - 0.8 \times 0) - (0.1)(0 \times -0.2 - 0.8 \times 0.3) + (0.2)(0 \times 0 - 0.6 \times 0.3)\)
• Simplified, \(\text{det}(A) = -0.048 + 0 + 0.036 = -0.012\)

Since the determinant is not zero, it confirms that the matrix has an inverse.

Cofactor Matrix

The cofactor matrix is a key step towards finding the inverse of a matrix. A cofactor is calculated for each element of the original matrix.

• For each element \(a_{ij}\) of the matrix, its minor is the determinant of the \((n-1)\times(n-1)\) matrix formed by deleting the \(i\)-th row and \(j\)-th column.
• The cofactor is given by \(C_{ij} = (-1)^{i+j}M_{ij}\), where \(M_{ij}\) is the minor of \(a_{ij}\).

Once all the cofactors are calculated, these are placed back into a cofactor matrix. This matrix is crucial to forming the adjugate matrix.

Adjugate Matrix

The adjugate matrix, also known as adjoint, is created by transposing the cofactor matrix. The process is straightforward:

• First, calculate the cofactor for each element of the original matrix.
• Arrange these cofactors into a matrix, called the cofactor matrix.
• Then, transpose this cofactor matrix (swap rows and columns) to form the adjugate matrix.

The adjugate matrix plays an essential role in calculating the actual inverse of the matrix using the inverse formula.

Inverse Formula

The inverse of a matrix \(A\) can be found using the formula \(A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)\), provided the determinant is nonzero.

• Step 1: Calculate the determinant of \(A\) as discussed earlier.
• Step 2: Find the adjugate matrix by transposing the cofactor matrix.
• Step 3: Use these in the inverse formula.

In our example, we've calculated the determinant to be \(-0.012\) and formed the adjugate matrix. Thus, the inverse of matrix \(A\) is given by:\[A^{-1} = \frac{1}{-0.012} \times \text{adj}(A)\]This results in a new matrix, which is the inverse of \(A\). This matrix is crucial for solving systems of equations where matrix \(A\) is involved.