Problem 69 - College Algebra

Question

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rrrr}7 & -3 & 0 & 2 \\\\-2 & 1 & 0 & -1 \\\4 & 0 & 1 & -2 \\\\-1 & 1 & 0 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

Use a graphing utility to compute the multiplicative inverse. This process gives you another matrix B. Verify B is the inverse by multiplying the original matrix with B and checking that the result is an identity matrix.

Step 1: Matrix Inversion using a Graphing Utility

To find the multiplicative inverse, one can use a graphing utility program capable of performing matrix operations. The matrix to input in this case is: \[\begin{bmatrix}7 & -3 & 0 & 2 \-2 & 1 & 0 & -1 \4 & 0 & 1 & -2 \-1 & 1 & 0 & -1\end{bmatrix}\]Once the matrix is entered, the utility can compute the multiplicative inverse. Let's call it matrix B.

Step 2: Verifying the Multiplicative Inverse

A matrix is the multiplicative inverse of another if their product is the identity matrix. To verify, multiply the original matrix (Matrix A) with the obtained inverse (Matrix B) in a graphing utility. If B is indeed the inverse, the result should be a 4x4 identity matrix, i.e., \[\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{array}\right]\]

Key Concepts

Graphing Utility

A graphing utility is a powerful tool designed to handle complex mathematical operations, including matrix calculations. It functions like a sophisticated calculator that can process matrices of various sizes, perform matrix multiplication, find determinants, and even calculate inverses. Given the complexity of such tasks, especially for larger matrices, this tool aids in simplifying these processes.
Using a graphing utility to find a matrix's multiplicative inverse involves entering the matrix entries into the tool. Once done, the utility can compute and display the inverse quickly.

• Enter the matrix by specifying the number of rows and columns.
• Input each element into the tool.
• Command the utility to calculate the inverse.

This pragmatic approach can save a significant amount of time compared to manual calculations, reducing potential errors in computations. A typical graphing utility will provide an easy interface for entering data and executing the operation.

Multiplicative Inverse

The multiplicative inverse of a matrix, often simply called the inverse, is central to numerous applications in mathematics and related fields. It exists for a matrix if certain conditions are met, notably if the matrix is square (same number of rows and columns) and its determinant is non-zero.
When a matrix has an inverse, it is denoted by the same letter, often with a superscript of -1, such as \( A^{-1} \). The concept is similar to the regular number inverse; for example, the inverse of 2 is 1/2.When you multiply a matrix by its inverse, the result is the identity matrix, the matrix equivalence of 1 in regular arithmetic. Remember:

• The matrix times its inverse equals the identity matrix.
• The inverse matrix effectively "undoes" the original matrix.
• Not all matrices have inverses (only those with a non-zero determinant).

Understanding when and how to use the inverse is crucial for solving systems of linear equations and other advanced mathematical problems.

Identity Matrix

An identity matrix is a special type of square matrix, in which all the elements of the principal diagonal are ones, and all other elements are zeros. It is pivotal in matrix multiplication as it serves as the "1" of matrices.

• A 2x2 identity matrix looks like this: \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
• A 3x3 identity matrix: \[ \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]
• For any square matrix \( A \), the equation \( A \times I = A \) holds, where \( I \) is the identity matrix of compatible size.

The identity matrix is crucial in proving that a given matrix \( B \) is indeed the inverse of another matrix \( A \), as \( A \times B = I \). Whenever you multiply a matrix by its identity matrix, the original matrix remains unchanged, a property that is integral in various calculations.

Matrix Operations

Matrix operations form the core of many mathematical computations across various applications, including linear algebra, computer graphics, and more. These operations include addition, subtraction, and, most significantly, multiplication.
Matrix multiplication, a key operation, involves combining two matrices to produce a third matrix. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Unlike normal multiplication, matrix multiplication is not commutative, meaning that \( AB eq BA \) in general.

• Matrix addition and subtraction require matrices to be of the same dimensions.
• Matrix multiplication can change the dimensions of the resulting matrix.
• Determinants, inverses, and transpositions are advanced matrix operations used frequently in solving complex problems.

These operations are pivotal in understanding how matrices interact and are manipulated, making them foundational knowledge for anyone working with linear algebra. Mastering these can broaden your capacity to manage and interpret matrices effectively.