Problem 36 - 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). $$ \left[\begin{array}{rrrr} 1 & 3 & -2 & 0 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & -2 & 1 \\ 0 & 0 & 0 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix exists as its determinant is non-zero. The inverse can be found using a graphing utility.

Step 1: Verify if the Matrix is Non-Singular

Analyze the given matrix. It's a 4x4 square matrix and of a special type, an upper triangular, where all entries below the main diagonal are zero. This means that its determinant can be easily calculated as the product of its diagonal elements. Calculate the determinant: det = 1*2*-2*5 = -20. The determinant is not equal to zero, which means that the matrix is non-singular and thus has an inverse.

Step 2: Use a Graphing Utility to find the Inverse

Enter the given matrix into a graphing utility and apply the inverse command. Most graphing calculators have a built-in function to calculate the inverse of a matrix. Specific steps can differ depending on the graphing utility used, so be sure to consult its manual or online support if necessary.

Step 3: Interpret the Result

The graphing utility will return a 4x4 matrix, which is the inverse of our given matrix. This matrix will be the final answer for our problem.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix. It plays a crucial role in linear algebra, particularly in determining whether a matrix has an inverse. For a matrix to have an inverse, it must be non-singular, which means its determinant is not zero.

In the given exercise, the matrix is upper triangular. This makes computing the determinant straightforward: just multiply the numbers along the main diagonal. For our matrix, this involves the values 1, 2, -2, and 5. When we calculate, we get:

• Multiply: 1 * 2 * -2 * 5 = -20

The result, -20, is not equal to zero, confirming that our matrix has an inverse. Understanding determinants simplifies checking for matrix invertibility.

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. This property significantly simplifies many computations, including finding the determinant.

In our scenario, the matrix provided is upper triangular. This means we only look at the diagonal elements when computing the determinant – this saves a lot of time compared to other matrices. Upper triangular matrices also have the interesting property that their inverses, if they exist, will also be upper triangular. However, computing an inverse directly still requires methodical steps or a tool like a graphing utility.

This matrix type is beneficial because it often leads to straightforward solutions in mathematical problems.

Graphing Utility

A graphing utility or graphing calculator is a tool that can handle a wide array of mathematical operations, including matrices and their inverses. Using a graphing utility to calculate the inverse of a matrix can be much simpler than manual computation, especially for larger matrices.

In this exercise, after determining the matrix has an inverse, we input the matrix into the graphing utility. While different tools have varied interfaces, most will have a function like "matrix inverse" to complete this task. The utility performs the complex calculations needed to find the inverse, providing results in moments.

Such technology is invaluable for students, aiding in verifying manual calculations and enhancing understanding of mathematical concepts through practice.