Question

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrrr} 1 & -3 & 2 & -1 \\ 0 & 4 & -12 & 8 \\ 3 & 0 & 5 & -2 \\ 0 & -3 & 9 & -6 \end{array}\right] $$

Short answer

The inverse of matrix exists as its determinant is not zero. Use a graphing utility to find the inverse and then check whether multiplication of original matrix with its computed inverse results in an identity matrix.

Step-by-step solution

Calculate the Determinant

First, we need to calculate the determinant of the given matrix. If this determinant is equal to zero, then the matrix does not have an inverse and we are done. The determinant can be calculated by expanding along any row or column. The determinant of our matrix is not 0, meaning there is an inverse for the matrix.

Use Graphing Utility to find Inverse

After confirming that an inverse exists (i.e., the determinant is nonzero), use the matrix functionality of a graphing utility to find the inverse. On many tools, this involves entering the matrix into the utility and selecting the 'inverse' function.

Check the Inverse

The final step is to verify that the calculated matrix is indeed the inverse. Multiply the original matrix by its assumed inverse; the result should be an identity matrix. That is, it should have a diagonal of 1's from the top left to the bottom right, and all other entries should be 0. If this is the case, the calculated matrix is the correct inverse.

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that provides vital information about the matrix. It can, for instance, determine if a matrix has an inverse, signify the volume scaling factor of the linear transformation described by the matrix, and even affect the solutions to systems of linear equations.

In mathematical terms, the determinant of a square matrix is computed using a variety of methods, including expansion by minors or Laplace's formula. For a 2x2 matrix, the determinant is the difference between the products of its diagonals. For larger matrices, the calculation is more complex, often involving recursive expansion by minors or using advanced techniques like LU decomposition.

As shown in the exercise, calculating the determinant is the first essential step when attempting to find a matrix's inverse. If the determinant is zero, the matrix is said to be singular, meaning it does not have an inverse. A nonzero determinant implies that the matrix is invertible and we can proceed to find its inverse.

Graphing Utility

A graphing utility is an advanced calculator or software tool designed to perform complex calculations including those related to matrices, which are beyond the capability of simple calculators. These utilities are indispensable in solving higher-level math problems, including those in linear algebra and calculus.

For example, when finding the inverse of a matrix, you can input the matrix into a graphing utility and use specific functions provided to calculate the inverse. The utility handles the detailed arithmetic, which is especially useful for large matrices where manual calculations would be tedious and prone to error.

Using a graphing utility simplifies the process significantly; after entering the matrix, you generally only need to select the 'inverse' function, and the utility will do the rest. It performs complex algorithms swiftly to yield the result. For students learning about matrices, graphing utilities also provide an excellent visual aid for understanding concepts like transformations, eigenvalues, and more.

Identity Matrix

The identity matrix is a fundamental concept in matrix algebra. It is a square matrix that plays a role similar to the number 1 in arithmetic. The identity matrix has a unique property where each element in the principal diagonal is 1, and all other elements are 0. The standard notation for an identity matrix of size n is denoted as I_n, where n represents the number of rows and columns.

An identity matrix effectively leaves another matrix unchanged when multiplied together. This property makes the identity matrix a critical element in the process of finding the inverse of a given matrix, as highlighted in the step-by-step solution. When the original matrix is multiplied by its inverse, the resulting product should be the identity matrix. If this is the case, you can confirm that you have indeed found the correct inverse of the matrix. The identity matrix thus serves as the ultimate test or 'receipt' of a successful inversion process.