Problem 4 - A Graphical Approach to College Algebra

Question

Determine whether A and B are inverses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] ; B=\left[\begin{array}{rr} 2 & 1 \\ -3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

A and B are not inverses because their products AB and BA do not equal the identity matrix.

Step 1: Calculate Matrix Product AB

To determine if matrices \( A \) and \( B \) are inverses, we first compute the product \( AB \). The matrices are given as \( A = \begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 1 \ -3 & 2 \end{bmatrix} \).To multiply the matrices \( AB \):1. Compute the element in the first row and first column: \( (2)(2) + (1)(-3) = 4 - 3 = 1 \).2. Compute the element in the first row and second column: \( (2)(1) + (1)(2) = 2 + 2 = 4 \).3. Compute the element in the second row and first column: \( (3)(2) + (2)(-3) = 6 - 6 = 0 \).4. Compute the element in the second row and second column: \( (3)(1) + (2)(2) = 3 + 4 = 7 \).Thus, \( AB = \begin{bmatrix} 1 & 4 \ 0 & 7 \end{bmatrix} \).

Step 2: Calculate Matrix Product BA

Next, we compute the product \( BA \):Use the given matrices \( B = \begin{bmatrix} 2 & 1 \ -3 & 2 \end{bmatrix} \) and \( A = \begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix} \).1. Compute the element in the first row and first column: \( (2)(2) + (1)(3) = 4 + 3 = 7 \).2. Compute the element in the first row and second column: \( (2)(1) + (1)(2) = 2 + 2 = 4 \).3. Compute the element in the second row and first column: \( (-3)(2) + (2)(3) = -6 + 6 = 0 \).4. Compute the element in the second row and second column: \( (-3)(1) + (2)(2) = -3 + 4 = 1 \).Thus, \( BA = \begin{bmatrix} 7 & 4 \ 0 & 1 \end{bmatrix} \).

Step 3: Determine if A and B are Inverses

Finally, recall that two matrices are inverses of each other if their product in both directions is the identity matrix, which is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).- The product \( AB = \begin{bmatrix} 1 & 4 \ 0 & 7 \end{bmatrix} \) is not the identity matrix.- Similarly, the product \( BA = \begin{bmatrix} 7 & 4 \ 0 & 1 \end{bmatrix} \) is not the identity matrix.Since neither \( AB \) nor \( BA \) is the identity matrix, matrices \( A \) and \( B \) are not inverses.

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It involves the process of combining two matrices to produce a third matrix. To multiply two matrices, we follow specific rules:

• Each element in the resulting matrix is computed by taking the dot product of the rows of the first matrix and the columns of the second matrix.
• An element in position (i, j) of the resulting matrix is obtained by multiplying the corresponding rows and columns and then adding their products.
• The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

To illustrate, consider matrices A and B. If matrix A is of size 2x2 and matrix B is also 2x2, they can be multiplied to produce a matrix of the same size (2x2). These matrices are multiplied step-by-step, focusing on row and column combinations, which emphasizes understanding over rote computation.
This structured approach helps ensure calculations are precise and helps students better grasp the concept of matrix multiplication.

Identity Matrix

The identity matrix is a fundamental concept in the study of matrices, acting as the '1' of matrix arithmetic. It plays a crucial role because it acts as the neutral element in matrix multiplication.

• For a square matrix, the identity matrix is a matrix with ones on the diagonal and zeros elsewhere. For example, a 2x2 identity matrix is represented as \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).
• When you multiply any matrix by the identity matrix (on either side), the original matrix remains unchanged.
• The identity property holds true only when the identity matrix and the matrix being multiplied are of compatible sizes.

The concept of the identity matrix is essential, especially when determining if two matrices are inverses. If the product of two matrices results in the identity matrix, it indicates that they are inverses of each other. This characteristic is key in theoretical and practical applications in matrix algebra.

Determinants

Determinants are numerical values derived from square matrices and are pivotal in defining important matrix properties such as invertibility. They provide insights into whether a matrix can be inverted or not.

• For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as: \( ad - bc \).
• A matrix can only have an inverse if its determinant is non-zero. A determinant of zero indicates the matrix is singular, which means it does not have an inverse.
• Determinants can also help assess the scalability of transformations represented by matrices in geometry and other fields.

Understanding determinants bridges the gap between theoretical matrix concepts and practical computations, allowing students to predict matrix behavior in equations and systems. They serve as a foundational step in problems related to matrix inverses and are useful for simplifying complex matrix algebra.