Problem 10 - College Algebra with Corequisite Support

Question

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 1 & 1 \end{array}\right], \quad B=\frac{1}{2}\left[\begin{array}{ccc} 2 & 1 & -1 \\ 0 & 1 & 1 \\ 0 & -1 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix \( A \) is the inverse of matrix \( B \) since both \( AB \) and \( BA \) equal the identity matrix.

Step 1: Verify Inverse by Multiplying A and B

To show matrix \( A \) is the inverse of matrix \( B \), first calculate the product \( AB \). Multiply each row of \( A \) by each column of \( B \) to get: \[AB = \begin{bmatrix}1 \times 1 + 0 \times 0 + 1 \times 0 & 1 \times \frac{1}{2} + 0 \times \frac{1}{2} - 1 \times \frac{1}{2} & 1 \times -\frac{1}{2} + 0 \times \frac{1}{2} + 1 \times \frac{1}{2} \0 \times 1 + 1 \times 0 - 1 \times 0 & 0 \times \frac{1}{2} + 1 \times \frac{1}{2} + 1 \times \frac{1}{2} & 0 \times -\frac{1}{2} + 1 \times \frac{1}{2} - 1 \times \frac{1}{2} \0 \times 1 + 1 \times 0 + 1 \times 0 & 0 \times \frac{1}{2} + 1 \times \frac{1}{2} + 1 \times \frac{1}{2} & 0 \times -\frac{1}{2} + 1 \times \frac{1}{2} + 1 \times \frac{1}{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{bmatrix}\]The product is the identity matrix.

Step 2: Verify Inverse by Multiplying B and A

Next, calculate the product \( BA \) by multiplying each row of \( B \) with each column of \( A \). The computation is as follows:\[BA = \begin{bmatrix}1 \times 1 + \frac{1}{2} \times 0 - \frac{1}{2} \times 0 & 1 \times 0 + \frac{1}{2} \times 1 - \frac{1}{2} \times 1 & 1 \times 1 + \frac{1}{2} \times -1 + \frac{1}{2} \times 1 \0 \times 1 + \frac{1}{2} \times 0 + \frac{1}{2} \times 0 & 0 \times 0 + \frac{1}{2} \times 1 + \frac{1}{2} \times 1 & 0 \times 1 + \frac{1}{2} \times -1 + \frac{1}{2} \times 1 \0 \times 1 - \frac{1}{2} \times 0 + \frac{1}{2} \times 0 & 0 \times 0 - \frac{1}{2} \times 1 + \frac{1}{2} \times 1 & 0 \times 1 - \frac{1}{2} \times -1 + \frac{1}{2} \times 1 \end{bmatrix} \]After calculation:\[BA = \begin{bmatrix} 1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{bmatrix}\]This product is also the identity matrix.

Step 3: Conclusion and Verification

Since both products \( AB \) and \( BA \) give the identity matrix, it confirms that the two matrices are inverses of each other. Thus, matrix \( A \) is indeed the inverse of matrix \( B \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra essential for understanding concepts like finding the inverse of a matrix. Unlike addition or subtraction, matrix multiplication involves multiplying rows by columns. The result is a new matrix with a specific arrangement of numbers. To multiply two matrices:

• Each element in the resulting matrix is the sum of the products of elements from the rows of the first matrix and the columns of the second matrix.
• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The result is a matrix whose dimensions are the outer dimensions of the two matrices being multiplied.

For instance, to show that two matrices are inverses, the product of matrix \(A\) and matrix \(B\) must equal the identity matrix. The process was demonstrated in the original exercise by carefully computing each element of the new matrix from the contributions of both \(A\) and \(B\). When multiplied, the matrices \(A\) and \(B\) produced an identity matrix, confirming their inverse relationship.

Identity Matrix

An identity matrix in the realm of linear algebra functions similarly to the number 1 in multiplication. This special type of matrix keeps a matrix unchanged, much like multiplying any number by 1 results in the original number. The identity matrix has ones on the diagonal (from top left to bottom right) and zeroes elsewhere. Some key points include:

• If multiplying a matrix by the identity matrix, the original matrix remains unchanged.
• The identity matrix acts as a neutral element in matrix multiplication.
• When two matrices are multiplied and give an identity matrix, they are inverses of each other.

In the exercise, the matrices \(AB\) and \(BA\) both equated to the identity matrix \(\begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}\), illustrating that matrix \(A\) is the inverse of matrix \(B\). This property is essential in solving systems of linear equations and simplifying complex mathematical equations.

College Algebra

College algebra covers a wide range of mathematical concepts including the study of matrices, which are stapled features in systems of equations, transformations, and linear algebra. Understanding how to work with matrices, including multiplication and finding inverses, is crucial. Here are some points to remember:

• Matrix operations are foundational topics that provide the tools for more advanced algebraic studies.
• Learning the functionality and properties of matrices aids students in solving applied math problems across scientific fields.
• The techniques are widely used in fields like physics, computer science, and economics for modeling and problem-solving.

The original exercise about matrix inverses is a typical exploration in college algebra, helping students solidify their grasp of theoretical concepts through practical computation. By practicing, they gain a comprehensive understanding of how matrices work and their vast applications.