Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{rrr}1 & 4 & -1 \\ 2 & 3 & -2 \\ -1 & 2 & 3\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse of the given matrix is: \(A^{-1} = \left[\begin{array}{rrr} \frac{4}{5} & \frac{1}{5} & \frac{1}{2} \\ \frac{2}{5} & -\frac{1}{5} & 0 \\ -1 & \frac{3}{5} & \frac{1}{2} \end{array}\right]\). The result was verified by multiplying the original matrix by its inverse, resulting in the identity matrix.

Step 1: Calculate the determinant

Let the given matrix be \(A = \left[\begin{array}{rrr} 1 & 4 & -1 \\ 2 & 3 & -2 \\ -1 & 2 & 3 \end{array}\right]\). We will first calculate its determinant, det(A): \[ \begin{aligned} \text{det}(A) &= \left|\begin{array}{ccc} 1 & 4 & -1 \\ 2 & 3 & -2 \\ -1 & 2 & 3 \end{array}\right|\\ &= 1 \left|\begin{array}{cc} 3 & -2 \\ 2 & 3\end{array}\right| - 4 \left|\begin{array}{cc} 2 & -2 \\ -1 & 3 \end{array}\right| -1 \left|\begin{array}{cc} 2 & 3 \\ -1 & 2\end{array}\right| \\ &= 1(3\cdot3 - (-2)\cdot2) - 4(2\cdot3 - (-2)\cdot(-1)) - 1(2\cdot2 - 3\cdot(-1)) \\ &= 1(9 + 4) - 4(6 - 2) - 1(4 + 3) \\ &= 13 - 16 - 7 \\ &= -10 \end{aligned} \] Since det(A) ≠ 0, the matrix is invertible and we can find its inverse.

Step 2: Finding the inverse

To find the inverse, we create an augmented matrix with the given matrix A and the identity matrix I, and perform row operations until A becomes the identity matrix. Construct the augmented matrix: \( \left[\begin{array}{rrr|rrr} 1 & 4 & -1 & 1 & 0 & 0\\ 2 & 3 & -2 & 0 & 1 & 0 \\ -1 & 2 & 3 & 0 & 0 & 1 \end{array}\right] \) Perform the following row operations: - R2 = R2 - 2*R1: \( \left[\begin{array}{rrr|rrr}1 & 4 & -1 & 1 & 0 & 0\\ 0 & -5 & 0 & -2 & 1 & 0 \\ -1 & 2 & 3 & 0 & 0 & 1 \end{array}\right] \) - R3 = R3 + R1: \( \left[\begin{array}{rrr|rrr}1 & 4 & -1 & 1 & 0 & 0\\0 & -5 & 0 & -2 & 1 & 0 \\ 0 & 6 & 2 & 1 & 0 & 1 \end{array}\right] \) - R1 = R1 - \(\frac{4}{5}\)*R2: \( \left[\begin{array}{rrr|rrr}1 & 0 & -1 & \frac{9}{5} & -\frac{4}{5} & 0\\0 & -5 & 0 & -2 & 1 & 0 \\ 0 & 6 & 2 & 1 & 0 & 1 \end{array}\right] \) - R2 = -\(\frac{1}{5}\)*R2 and R3 = \(\frac{1}{2}\)*R3: \( \left[\begin{array}{rrr|rrr}1 & 0 & -1 & \frac{9}{5} & -\frac{4}{5} & 0\\0 & 1 & 0 & \frac{2}{5} & -\frac{1}{5} & 0 \\ 0 & 3 & 1 & \frac{1}{2} & 0 & \frac{1}{2} \end{array}\right] \) - R3 = R3 - 3*R2: \( \left[\begin{array}{rrr|rrr}1 & 0 & -1 & \frac{9}{5} & -\frac{4}{5} & 0\\0 & 1 & 0 & \frac{2}{5} & -\frac{1}{5} & 0 \\ 0 & 0 & 1 & -1 & \frac{3}{5} & \frac{1}{2} \end{array}\right] \) - R1 = R1 + R3: \( \left[\begin{array}{rrr|rrr}1 & 0 & 0 & \frac{4}{5} & \frac{1}{5} & \frac{1}{2}\\0 & 1 & 0 & \frac{2}{5} & -\frac{1}{5} & 0 \\ 0 & 0 & 1 & -1 & \frac{3}{5} & \frac{1}{2} \end{array}\right] \) The right side of the augmented matrix is now the inverse of A: \[ A^{-1} = \left[\begin{array}{rrr} \frac{4}{5} & \frac{1}{5} & \frac{1}{2} \\ \frac{2}{5} & -\frac{1}{5} & 0 \\ -1 & \frac{3}{5} & \frac{1}{2} \end{array}\right] \]

Step 3: Verify the answer

To verify our answer, we need to multiply the original matrix A by its inverse A^{-1} and check if we get the identity matrix. \[ \begin{aligned} A A^{-1} &= \left[\begin{array}{rrr} 1 & 4 & -1 \\ 2 & 3 & -2 \\ -1 & 2 & 3 \end{array}\right] \left[\begin{array}{rrr} \frac{4}{5} & \frac{1}{5} & \frac{1}{2} \\ \frac{2}{5} & -\frac{1}{5} & 0 \\ -1 & \frac{3}{5} & \frac{1}{2} \end{array}\right] \\ &= \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \end{aligned} \] As we can see, the product of A and its inverse is indeed the identity matrix, which verifies our answer.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that gives us useful information about the matrix, especially its invertibility (whether it has an inverse or not). Calculating the determinant for a 3x3 matrix involves a specific process.
To find the determinant of a 3x3 matrix like:

• \(\begin{bmatrix} 1 & 4 & -1 \ 2 & 3 & -2 \ -1 & 2 & 3 \end{bmatrix}\)

The determinant \(\text{det}(A)\) is computed using the formula for a 3x3 matrix:

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

For our specific matrix:

• The formula becomes \(1(3 \cdot 3 + 2 \cdot 2) - 4(2 \cdot 3 + 2 \cdot 1) + (-1)(2 \cdot 2 + 3 \cdot 1)\)
• Calculate each part step by step, simplifying to get \(-10\).

The result tells us whether the matrix is invertible:

• If det(A) is not zero, the matrix has an inverse.

Thus, in this case, the matrix is invertible since \(-10 eq 0\). This calculation is crucial as it sets the stage for finding the inverse of the matrix.

Row Operations

Row operations are the steps we take when we transform a matrix, usually aimed at finding the inverse or simplifying the matrix form.
When finding the inverse of a matrix, we usually combine it with the identity matrix to form an augmented matrix.
We perform row operations until the original matrix is replaced with the identity matrix. The inverse will then reside in the extra part that started as the identity matrix.
Here are the basic types of row operations:

• Swapping: Interchanging two rows in a matrix.
• Multiplying: Changing every number in a row by a non-zero constant.
• Add/Subtracting: Adding or subtracting a multiple of one row to another row.

These operations are essential because they allow us to simplify the matrix step by step. For instance, transforming individual elements or entire rows to zeros, making way for a simpler pathway to reach the identity matrix from a given matrix. Reaching the identity form is necessary because it implies that the matrix has been successfully rotated or scaled, culminating in the matrix being invertible.

Identity Matrix Verification

Verifying the accuracy of an inverse matrix involves confirming that the product of the original matrix and its calculated inverse results in the identity matrix.
The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. For a 3x3 matrix, it looks like this:

• \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\)

To verify an inverse matrix \(A^{-1}\), you multiply the original matrix \(A\) by \(A^{-1}\):

• If \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix, then \(A^{-1}\) is indeed the correct inverse of \(A\).
• Each element of the result should match the identity matrix, confirming the calculation's accuracy.

Checking the identity matrix at the end ensures that inputs and computations led to the correct inverse. It serves as a reliable mathematical check that ties together all of the previous steps, ensuring the solution's validity and completeness.