Problem 52 - A Graphical Approach to College Algebra

Question

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} 2 & 1 & -1 \\ 0 & 2 & 1 \\ 3 & 2 & -1 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 0 \\ 2 & -1 \\ 3 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

\(AB = \begin{bmatrix} 1 & -2 \\ 7 & -1 \\ 4 & -3 \end{bmatrix}\). \(BA\) is not possible.

Step 1: Confirm Compatibility for Matrix Multiplication

First, verify if matrices \(A\) and \(B\) can be multiplied. The number of columns in the first matrix must match the number of rows in the second matrix. Matrix \(A\) is \(3 \times 3\) and matrix \(B\) is \(3 \times 2\). The multiplication \(AB\) is possible because both have the same number of rows and columns (3). Therefore, \(AB\) results in a \(3 \times 2\) matrix. Now check if \(BA\) is possible: \(B\) is \(3 \times 2\) and \(A\) is \(3 \times 3\), which is not compatible (2 columns in \(B\), 3 rows in \(A\)). So, \(BA\) is not possible.

Step 2: Multiply Matrices to Find AB

To find \(AB\), multiply each element of the rows of matrix \(A\) by the corresponding elements of the columns of matrix \(B\) and sum the products.1. First row of \(AB\):\[\begin{bmatrix} 2 & 1 & -1 \end{bmatrix} \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} = (2)(1) + (1)(2) + (-1)(3) = 2 + 2 - 3 = 1\]\[\begin{bmatrix} 2 & 1 & -1 \end{bmatrix} \begin{bmatrix} 0 \ -1 \ 1 \end{bmatrix} = (2)(0) + (1)(-1) + (-1)(1) = 0 - 1 - 1 = -2\]2. Second row of \(AB\):\[\begin{bmatrix} 0 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} = (0)(1) + (2)(2) + (1)(3) = 0 + 4 + 3 = 7\]\[\begin{bmatrix} 0 & 2 & 1 \end{bmatrix} \begin{bmatrix} 0 \ -1 \ 1 \end{bmatrix} = (0)(0) + (2)(-1) + (1)(1) = 0 - 2 + 1 = -1\]3. Third row of \(AB\):\[\begin{bmatrix} 3 & 2 & -1 \end{bmatrix} \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} = (3)(1) + (2)(2) + (-1)(3) = 3 + 4 - 3 = 4\]\[\begin{bmatrix} 3 & 2 & -1 \end{bmatrix} \begin{bmatrix} 0 \ -1 \ 1 \end{bmatrix} = (3)(0) + (2)(-1) + (-1)(1) = 0 - 2 - 1 = -3\]Thus, \(AB = \begin{bmatrix} 1 & -2 \ 7 & -1 \ 4 & -3 \end{bmatrix}\).

Key Concepts

Matrix Compatibility

Matrix compatibility is crucial when multiplying two matrices. To determine if two matrices can be multiplied, you need to check the dimensions. The number of columns in the first matrix must match the number of rows in the second.

For example, consider matrices \(A\) and \(B\) from the exercise. Matrix \(A\) is \(3 \times 3\), meaning it has 3 rows and 3 columns. Matrix \(B\) is \(3 \times 2\), with 3 rows and 2 columns. Since the number of columns in \(A\) equals the number of rows in \(B\), they are compatible for multiplication as \(AB\).

However, for \(BA\) to be possible, \(B\) would need 3 columns (since \(A\) has 3 rows), but it only has 2. Therefore, \(BA\) is not compatible for multiplication.

Product of Matrices

The product of two matrices involves multiplying rows by columns. When matrices are compatible, the result is a new matrix with dimensions based on the number of rows in the first and the number of columns in the second.

From the exercise, multiplying \(A\) and \(B\) results in a \(3 \times 2\) matrix because:\[ \begin{align*} \text{(rows of A)} & \times \text{(columns of B)} \end{align*} \]
Here’s how you calculate each element of the result matrix:

• Take a row from matrix \(A\).
• Multiply each element by the corresponding element in a column of matrix \(B\).
• Sum all these products to find a single element of the result matrix.

This process is repeated for each row of \(A\) and each column of the resulting matrix. It's about turning rows into columns and summing them up.

Matrix Dimensions

Matrix dimensions provide essential information about the structure and size of a matrix. They are written as "rows by columns".

Understanding dimensions helps decide compatibility and predict the size of the resulting matrix after multiplication. For instance:

• Matrix \(A\) is \(3 \times 3\), meaning it has 3 rows and 3 columns.
• Matrix \(B\) is \(3 \times 2\), having 3 rows and 2 columns.

These dimensions play a key role in the multiplication process. When multiplying, the resulting matrix will have dimensions resulting from the number of rows in the first matrix and the number of columns in the second.

In our example, since \(AB\) is possible, the resulting matrix is \(3 \times 2\), which gives us a clear picture of the outcome.