Problem 65 - Algebra and Trigonometry

Question

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{rr} {3} & {-1} \\ {-2} & {1} \end{array}\right] $$

Step-by-Step Solution

Answer

To find the multiplicative inverse of the matrix, input the matrix into a graphing utility or calculator capable of matrix operations and use the function to calculate the inverse. Then, multiply the resulting matrix by the original to check if the output is the identity matrix. If it is, then the solution is correct.

Step 1: Calculate the Inverse of the Matrix

First, use a graphing utility or calculator that has the capability to compute the inverse of a 2x2 matrix. Input the matrix into the calculator in the following format: \[ \begin{pmatrix} 3 & -1 \ -2 & 1 \end{pmatrix} \]After entering this matrix, use the calculator function to compute the inverse. The result should be the inverse of the given matrix.

Step 2: Checking the correctness of the Inverse

The result gotten from the calculator is believed to be the inverse of the matrix. However, an inverse of a matrix when multiplied by the matrix should give an identity matrix. To check the correctness of the result, multiply the original matrix with the matrix obtained as its inverse:\[\begin{pmatrix} 3 & -1 \ -2 & 1 \end{pmatrix} *\begin{pmatrix} a & b \ c & d \end{pmatrix}\] If the product equals the identity matrix\[\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]then the inverse calculated is correct.

Key Concepts

Matrix Inverse

Understanding the concept of matrix inverses is crucial in linear algebra and has broad applications in fields such as computer graphics, optimization, and cryptography. Likened to the reciprocal of a number, the matrix inverse is a matrix that, when multiplied with the original matrix, results in the identity matrix.

Let's consider a matrix \( A \). The inverse of \( A \), denoted as \( A^{-1} \), must fulfill the condition \( A \times A^{-1} = A^{-1} \times A = I \), where \( I \) is the identity matrix. It's important to note that not all matrices have inverses, and only square matrices—those with the same number of rows and columns—can potentially have an inverse. A matrix without an inverse is called singular or non-invertible.

In practice, finding a matrix inverse requires the use of methods such as Gaussian elimination or the calculation of the adjugate and determinant for a 2x2 matrix. Remember, the ability to invert a matrix can be critical in solving systems of linear equations and in analyzing linear transformations.

Graphing Calculator Use

Graphing calculators are powerful tools that aid in complex mathematical computations, which includes finding the inverse of matrices. These calculators have built-in functions that allow for easy manipulation of matrices and can save a considerable amount of time compared to manual calculations.

To use a graphing calculator for finding a matrix inverse, follow these general steps:

• Enter the matrix into the calculator. This is often done by accessing the matrix mode or editor and inputting the elements row by row.
• Access the function that calculates the matrix inverse. The exact sequence of buttons may vary depending on the model of the calculator.
• Review the output, which should be the inverse of the matrix you entered.

Note that if the matrix is non-invertible, the calculator will typically return an error message indicating that the calculation cannot be completed. Always verify the accuracy of the calculator's output by performing the required multiplication check with the original matrix.

Identity Matrix

The identity matrix, denoted by \( I \), plays a pivotal role in matrix algebra. It is a square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. For any matrix \( A \), if it is multiplicable with the identity matrix, the result will be the matrix itself, i.e., \( A \times I = A \).

An identity matrix acts much like the number 1 in regular multiplication, hence it's often referred to as the multiplicative identity of matrices. The presence and use of the identity matrix are fundamental in operations such as finding matrix inverses and solving linear equations. In the verification process of a matrix inverse, the product of a matrix and its purported inverse should yield the identity matrix, which serves as a crucial correctness check.