Problem 21 - Algebra for College Students

Question

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{rr} 2 & -1 \\ -5 & 3 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & 1 \\ 5 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

Both \(AB\) and \(BA\) equal the identity matrix \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\).

Step 1: Understand Matrix Multiplication

Matrix multiplication involves taking the dot product of rows and columns. For matrices \(A\) and \(B\), the element in the first row and first column of the product matrix is the dot product of the first row of \(A\) and the first column of \(B\).

Step 2: Compute \(A \times B\)

To multiply matrix \(A\) by matrix \(B\), perform the following calculations:1. First row, first column: \(2 \times 3 + (-1) \times 5 = 6 - 5 = 1\).2. First row, second column: \(2 \times 1 + (-1) \times 2 = 2 - 2 = 0\).3. Second row, first column: \((-5) \times 3 + 3 \times 5 = -15 + 15 = 0\).4. Second row, second column: \((-5) \times 1 + 3 \times 2 = -5 + 6 = 1\).So, \(A \times B = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Step 3: Compute \(B \times A\)

To multiply matrix \(B\) by matrix \(A\), perform the following calculations:1. First row, first column: \(3 \times 2 + 1 \times (-5) = 6 - 5 = 1\).2. First row, second column: \(3 \times (-1) + 1 \times 3 = -3 + 3 = 0\).3. Second row, first column: \(5 \times 2 + 2 \times (-5) = 10 - 10 = 0\).4. Second row, second column: \(5 \times (-1) + 2 \times 3 = -5 + 6 = 1\).So, \(B \times A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Step 4: Verify Results

Both products \( A B \) and \( B A \) resulted in the identity matrix \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), confirming our calculations were performed correctly.

Key Concepts

Dot Product

In the realm of matrix multiplication, the "dot product" is an essential concept to grasp. To compute the dot product of two matrices, you focus on the rows of the first matrix and the columns of the second matrix. The calculation involves multiplying corresponding elements and then summing those products. For example, if you have the first row of matrix A and the first column of matrix B, the dot product involves:

• Multiplying each element in the row of A with the corresponding element in the column of B.
• Adding up all these results to get a single number.

Let's see this in action with the exercise example:For the element in the first row and first column of the resulting matrix, you calculate:\[(2 \times 3) + ((-1) \times 5) = 6 - 5 = 1\]This operation shows how each position in the resulting matrix is derived from a dot product between rows and columns.

Identity Matrix

The identity matrix can be thought of as the "1" in the world of matrices. It's a special matrix where all the elements on the main diagonal are 1, and all other elements are 0. Think of it like this: multiplying any matrix by the identity matrix leaves it unchanged, just like multiplying a number by 1.In our exercise, compute matrices \(AB\) and \(BA\), both results turn out to be the identity matrix \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). This surprising result is important because it confirms a unique relationship between the matrices A and B.

• Each diagonal element \(1\) ensures the existence of each row or column distinctly.
• Each off-diagonal element \(0\) means no interaction between other entries during multiplication.

Identity matrices serve as a neutral element in matrix multiplication, crucial for verifying the correctness of operations.

Matrix Operations

Matrix operations, including multiplication, involve calculating new matrices by combining two old ones. In contrast to simple arithmetic multiplication, matrix multiplication considers the structure of matrices—rows and columns—leading to a new matrix size.Here are some key rules:

• Matricies A (size \(m \times n\)) and B (size \(p \times q\)) can only be multiplied if the number of columns in A equals the number of rows in B, i.e., \(n = p\).
• The resulting matrix will have dimensions \(m \times q\).

In our example, both matrices A and B are 2x2, which suits the criteria for multiplication. Each element of the resulting matrix is formed by taking the dot product of rows from the first matrix with columns of the second matrix. Understanding these operations means being able to handle complex problems, manipulate data more flexibly, and represents an essential toolset in linear algebra and many applied disciplines.