Problem 29 - Precalculus: Mathematics for Calculus

Question

Use a graphing calcuIator to find the inverse of the matrix, if it exists. $$\left[\begin{array}{rrrr}-1 & -4 & 0 & 1 \\\1 & 0 & -1 & 0 \\\0 & 4 & 1 & -2 \\\2 & 2 & -2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is invertible, and its inverse is found using a graphing calculator.

Step 1: Check Matrix Invertibility

First, we need to check if the given matrix is invertible. A matrix is invertible if its determinant is non-zero. Use the graphing calculator to compute the determinant of the matrix:\[\text{det}(A) = \begin{vmatrix} -1 & -4 & 0 & 1 \ 1 & 0 & -1 & 0 \ 0 & 4 & 1 & -2 \ 2 & 2 & -2 & 0 \end{vmatrix}\]If the determinant is zero, the matrix is not invertible.

Step 2: Calculate the Determinant

Using the graphing calculator, calculate the determinant of the matrix. Enter the matrix into the calculator and find the determinant. If the result is a non-zero value, such as 10 or -5, the matrix is invertible. If the result is 0, the matrix is not invertible. For our given matrix, you find the determinant to be non-zero.

Step 3: Find the Inverse Using Calculator

Given that the determinant is non-zero, use the matrix function on the graphing calculator to find the inverse. Enter the matrix and use the 'inv' or inverse function to compute the inverse matrix.

Step 4: Verify the Inverse

To ensure the inverse matrix is correct, multiply the original matrix by the calculated inverse using the calculator. You should obtain the identity matrix:\[\begin{bmatrix} -1 & -4 & 0 & 1 \ 1 & 0 & -1 & 0 \ 0 & 4 & 1 & -2 \ 2 & 2 & -2 & 0 \end{bmatrix} \times \text{Inverse} = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}\]If it equals the identity matrix, the inverse is correct.

Key Concepts

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. This value can help us determine if a matrix is invertible. Calculating the determinant of a matrix involves complex operations, but graphing calculators simplify this task. For a 4x4 matrix like the one we're working with, manually computing the determinant would involve many steps and sub-determinants.
To find the determinant:

• Input the matrix into the calculator.
• Use the calculator's function that computes the determinant, usually labeled as 'det' or similar.

Once calculated, if the determinant is non-zero, it implies that the matrix could potentially have an inverse.
If it is zero, the matrix is not invertible. This non-zero determinant is an essential precursor to moving on to the next steps of matrix inversion.

Invertibility of Matrices

Invertibility refers to whether a matrix has an inverse—a sort of mathematical "opposite." A matrix must meet certain criteria, such as having a non-zero determinant, to be invertible.
Matrix inversion is a crucial concept in linear algebra because it allows us to solve equations represented in matrix form and perform various transformations. Here are the key steps to check for invertibility:

• Calculate the determinant of the matrix using a suitable method like a graphing calculator.
• Verify that the determinant is non-zero.

If the matrix passes this check, you can use computational tools, such as graphing calculators, to find its inverse. These calculators offer easy-to-use functions to perform inversion operations, taking care of the detailed calculations for you.

Graphing Calculator

Using a graphing calculator is a powerful way to perform matrix calculations, including finding determinants and inverses. These calculators can handle complex operations through built-in functions. For students, this means more efficient problem-solving without the hassle of manual computations.
Here's how to use a graphing calculator for matrix inversion:

• Input the entire matrix into the calculator's matrix function.
• Calculate the determinant to check invertibility.
• Use the 'inv' or inverse function to compute the inverse, if the determinant is non-zero.
• Verify the result by multiplying the original matrix by the inverse; this should yield the identity matrix.

This user-friendly tool has an interface that guides you through each step, confirming the accuracy of your results. By making use of such technology, students can better focus on understanding concepts rather than getting bogged down in complex calculations.