Problem 38 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}2 & -1 & -5 \\\4 & -1 & 6 \\\\-2 & 0 & 9 \end{array}\right], \quad B=\left[\begin{array}{rr}1 & 2 \\\\-1 & -1 \\\2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

AB is \([[-7, 5], [17, 9], [16, -4]]\), BA is not possible.

Step 1: Determine Compatibility of Matrix Multiplication

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(A\) is a 3x3 matrix and matrix \(B\) is a 3x2 matrix. Therefore, \(AB\) is possible because \(A\) has 3 columns and \(B\) has 3 rows. However, \(BA\) is not possible because \(B\) (2 columns) and \(A\) (3 rows) do not meet the criteria for multiplication.

Step 2: Multiply Matrices (A * B)

Now, we multiply matrices \(A\) and \(B\). The result will be a 3x2 matrix, with each element calculated as the dot product of the rows of \(A\) with the columns of \(B\).- First row: \\(2*1 + (-1)(-1) + (-5)*2 = 2 + 1 - 10 = -7\) \\(2*2 + (-1)*(-1) + (-5)*0 = 4 + 1 = 5\) \- Second row: \\(4*1 + (-1)*(-1) + 6*2 = 4 + 1 + 12 = 17\) \\(4*2 + (-1)*(-1) + 6*0 = 8 + 1 = 9\) \- Third row: \\((-2)*1 + 0*(-1) + 9*2 = -2 + 0 + 18 = 16\) \\((-2)*2 + 0*(-1) + 9*0 = -4 + 0 = -4\)The resulting matrix is: \\[\begin{bmatrix} -7 & 5 \ 17 & 9 \ 16 & -4 \end{bmatrix}\]

Step 3: Summarize Findings

Since matrix multiplication \(BA\) is not possible, our only result is the matrix obtained from \(AB\). The matrix \(AB\) is: \\[\begin{bmatrix} -7 & 5 \ 17 & 9 \ 16 & -4 \end{bmatrix}\] \Thus, the product \(AB\) exists and is as calculated above.

Key Concepts

Dot Product

The dot product is a fundamental operation in matrix multiplication, specifically when calculating the product of two matrices. When multiplying matrices, each element in the resulting matrix is derived from the dot product of a row from the first matrix and a column from the second matrix. This involves summing the products of corresponding elements in the row and column.

To illustrate, consider two matrices, where matrix \(A\) has dimensions \(3 \times 3\), and matrix \(B\) is \(3 \times 2\). To find the entry in the first row and first column of the resulting matrix \(AB\), you calculate the dot product of the first row of matrix \(A\) and the first column of matrix \(B\):

• Multiply each element of the row by the corresponding element of the column.
• Add these products together.

For example, given the first row of \(A\) as \([2, -1, -5]\) and the first column of \(B\) as \([1, -1, 2]\), the calculation is:

\(2 \cdot 1 + (-1) \cdot (-1) + (-5) \cdot 2 = 2 + 1 - 10 = -7\)

This result, \(-7\), is the element of the resulting matrix in the first row and first column.

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrix operations such as multiplication. Each matrix has a specific structure defined by its dimensions, expressed in terms of rows by columns (rows × columns).

For instance, the matrix \(A\) in the exercise is a \(3 \times 3\) matrix, meaning it has three rows and three columns. On the other hand, matrix \(B\) is a \(3 \times 2\) matrix, which contains three rows and two columns.

• The number of rows represents how many horizontal lines of elements there are.
• The number of columns indicates how many vertical lines of elements exist.

Matrix dimensions also determine the feasible operations that can be performed on them. Hence, knowing these dimensions is essential for operations such as matrix multiplication.

Compatibility of Multiplication

When discussing matrix multiplication, it is essential to understand the concept of compatibility. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

Given the matrices in this exercise, matrix \(A\) is \(3 \times 3\), and matrix \(B\) is \(3 \times 2\). This setup allows for the multiplication \(AB\), as the 3 columns in matrix \(A\) match the 3 rows in matrix \(B\).

However, the multiplication \(BA\) is not possible because matrix \(B\) has 2 columns, and matrix \(A\) has 3 rows. Therefore:

• For \(AB\): Works because the inner dimensions match (3 with 3).
• For \(BA\): Does not work due to a mismatch in inner dimensions (2 with 3).

Ensuring compatibility is the first step before performing any matrix multiplication. If this condition is not met, the multiplication cannot proceed.