Question

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 10 & 5 & -7 \\ -5 & 1 & 4 \\ 3 & 2 & -2 \end{array}\right] $$

Short answer

The inverse of the matrix cannot be calculated in this setting, as this requires a graphing utility. However, to find it, the matrix must be entered into the graphing utility, verified that it is indeed invertible, and then the utility's inverse function should be used to find the inverse.

Step-by-step solution

Enter the Matrix

Start by entering the following matrix into the graphing utility: \( \left[\begin{array}{rrr} 10 & 5 & -7 \ -5 & 1 & 4 \ 3 & 2 & -2 \end{array}\right] \). This will provide the baseline matrix off of which the inverse matrix will be calculated. Ensure it is entered correctly to avoid any miscalculations.

Check if the Matrix is Invertible

Before proceeding with the calculations of the inverse, it is essential to confirm that the matrix is indeed invertible. This can be done by calculating the determinant of the matrix and checking if it is zero. If the determinant is zero, the matrix does not have an inverse, and calculations should stop. Most graphing utilities will provide an error message if the determinant is zero.

Find the Inverse of the Matrix

If the matrix is invertible, proceed to find the inverse of the matrix. Depending on the utility being used, there are generally two ways to find the inverse. The first is by going into the matrix menu, selecting the matrix in question, and hitting the inverse key. This will usually provide the inverted matrix instantaneously. The other method of finding an inverse on a graphing utility includes typing the name of the matrix, then pressing the \(^{-1}\) button. This will also return the inverted matrix.

Key concepts

Matrix Invertibility

Understanding the concept of matrix invertibility is crucial for numerous applications in linear algebra, including solving systems of linear equations and performing transformations in computer graphics. A matrix is considered invertible or non-singular if there exists another matrix which, when multiplied by the original matrix, results in the identity matrix.

In practical terms, the identity matrix is akin to the number 1 in regular multiplication – it is the neutral element that leaves any matrix unchanged when multiplied by it. For a matrix to be invertible, it must be a square matrix, meaning it has the same number of rows and columns. Most importantly, as touched on in the step-by-step solution, the determinant of the matrix must not be zero. This single value, the determinant, acts as a gatekeeper; if it's zero, the matrix cannot be inverted, and if it's nonzero, we may proceed with finding its inverse.

To enhance understanding, consider why the determinant's non-zero condition is essential: imagine trying to divide by zero in arithmetic—it's impossible. Likewise, a zero determinant signifies there is no unique solution to the system of equations represented by the matrix—thus making the concept of an inverse nonsensical in this context.

Graphing Utility

A graphing utility is an invaluable tool for both students and professionals working with complex mathematical problems. The use of such a utility for matrices extends to its capabilities to handle computations that are otherwise tedious and prone to error when done by hand. When it comes to finding the inverse of a matrix, a graphing utility simplifies the process significantly. After correctly inputting the matrix, as outlined in the original exercise, the graphing utility performs the necessary calculations instantaneously, saving time and reducing the likelihood of manual errors.

For educational purposes, it's important to become familiar with the features of your graphing utility. Knowing how to enter matrices, access different functions in the matrix menu, and interpreting any error messages will boost confidence and competence in handling matrix-related questions. While the actual operations are performed within the utility, the understanding of concepts like matrix invertibility should always be at the forefront to validate the results provided by the tool.

Determinant of a Matrix

The determinant is a scalar value that carries a wealth of information about a matrix. It can tell us whether a matrix is invertible, it can give insight into the volume scaling factor of the linear transformation associated with the matrix, and it is utilized across various areas such as differential equations and economics. To calculate the determinant of a 2x2 matrix, for example, we subtract the product of the top right and bottom left elements from the product of the top left and bottom right elements. The process for a 3x3 matrix is more complex, involving the calculation of minors and cofactors, or employing the Sarrus rule.

As described in the solution provided, the determinant of our given matrix must not be zero for the matrix to have an inverse. This is because the determinant can be interpreted as a measure of how the matrix, when viewed as a transformation, distorts the space it operates in. If the determinant is zero, the space is squashed completely along some dimension, meaning the original configuration cannot be retrieved—thus, an inverse that would 'undo' the transformation does not exist. Grasping this concept provides a deeper appreciation for the significance of the determinant beyond just a number to check before finding an inverse.