Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 0.6 & 0 & -0.3 \\ 0.7 & -1 & 0.2 \\ 1 & 0 & -0.9 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix is: \[\left[\begin{array}{rrr}2.1429 & -0.2381 & -0.7143 \0 & 2.1429 & -1.4286 \-0.4762 & 0.2381 & -1.4286\end{array}\right]\]

Step 1: Check If the Matrix is Invertible

We are given a 3x3 matrix, so it is a square matrix. Next we need to compute the determinant of the matrix. If the determinant is not zero then the matrix is invertible. In this case, the determinant equals -0.42, which is not zero, so this matrix is invertible.

Step 2: Calcuate the Adjunct of Matrix

The adjunct of the matrix is the transpose of the cofactor matrix. Which in this case, it happens to be:\[\left[\begin{array}{rrr}-0.9 & 0.1 & 0.3 \0 & -0.9 & 0.6 \0.2 & -0.1 & 0.6 \end{array}\right]\]

Step 3: Calculate the Inverse of the Matrix

Now, to calculate the inverse of the matrix, divide each element in the adjoint matrix by the determinant of the original matrix. This yields:\[\left[\begin{array}{rrr}2.1429 & -0.2381 & -0.7143 \0 & 2.1429 & -1.4286 \-0.4762 & 0.2381 & -1.4286\end{array}\right]\] which is the inverse of the original matrix.

Key Concepts

Determinants

The determinant is a crucial value that can be calculated from a square matrix. It helps us understand various properties of the matrix, such as invertibility. To determine if a matrix can have an inverse, we calculate its determinant. If the determinant is zero, the matrix is singular and does not have an inverse. If it is non-zero, the matrix is invertible. For the given matrix, the determinant was calculated as -0.42, which means it is invertible.

Determinants are calculated using specific formulas that vary depending on the size of the matrix. For a 3x3 matrix, the determinant can be found by using a combination of its elements and their respective minors. This involves a bit of computation, but knowing how to find a determinant is fundamental for understanding matrix invertibility.

Cofactor Matrix

To develop an understanding of the cofactor matrix, we need to talk about minors and cofactors. Each element of the matrix has a minor, which is the determinant of the submatrix that remains after removing the element's row and column. A cofactor is simply the minor with a sign change, depending on the position \((i,j)\) within the matrix, following this pattern:

• C_ij = (-1)^{i+j} * M_ij

where C_ij is the cofactor and M_ij is the minor.

Once all the cofactors of a matrix are found, they form the cofactor matrix. This matrix is a key step for finding the inverse matrix because it leads us to the adjunct, which is instrumental in the inversion process.

Adjunct of a Matrix

The adjunct, often called the adjugate, of a matrix is a related concept essential for finding the inverse. This matrix is obtained by taking the cofactor matrix and transposing it. Transposition involves flipping the matrix over its diagonal, effectively swapping rows and columns. For example, if the cofactor matrix of \([[0.9, 0.1, 0.3], [0, 0.9, 0.6], [0.2, 0.1, 0.6]])\) is calculated, its transpose would rearrange the elements as:

• First row becomes the first column
• Second row becomes the second column
• Third row becomes the third column

The adjugate is crucial because it's used alongside the determinant to calculate the inverse of a matrix. Without transposing the cofactor matrix, the inverse calculation would not be accurate.

Inverse Matrix Calculation

Calculating the inverse of a matrix requires a few definitive steps. After confirming that the determinant is not zero, the next step involves using the adjunct (or adjugate) and the determinant.

For a matrix \( A \), the inverse is defined as \( A^{-1} = \text{adj}(A) \times \frac{1}{\text{det}(A)} \). This formula showcases how we can derive the inverse by scaling the adjunct matrix with the reciprocal of the determinant. Detailing the steps:

• Calculate the adjunct of the matrix
• Find the determinant of the original matrix
• Divide each element of the adjunct by the determinant
• Result is the inverse of the matrix

When applied to the given matrix, the inverse matrix was found successfully, showing that matrix inversion is systematic and relies heavily on both the adjunct and the determinant. Knowing how to perform these calculations ensures we can solve systems of linear equations effectively using inverse matrices.