Problem 34 - College Algebra with Modeling and Visualization

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 can be found and is \(\left[\begin{array}{cc}2 & -2 \\7 & -1 \\7 & -3\end{array}\right]\). BA is not possible.

Step 1: Understand the dimensions

First, determine the dimensions of matrices \(A\) and \(B\). Matrix \(A\) is a \(3 \times 3\) matrix, and matrix \(B\) is a \(3 \times 2\) matrix.

Step 2: Check the possibility of finding AB

Matrix multiplication is possible only if the number of columns in the first matrix equals the number of rows in the second matrix. For \(AB\), the multiplication is possible because \(A\) has 3 columns and \(B\) has 3 rows.

Step 3: Calculate AB

To find \(AB\), multiply each row of \(A\) by each column of \(B\). The result is a \(3 \times 2\) matrix. Perform the multiplication:\[AB = \left[\begin{array}{cc}2 \times 1 + 1 \times 2 + (-1) \times 3 & 2 \times 0 + 1 \times (-1) + (-1) \times 1 \0 \times 1 + 2 \times 2 + 1 \times 3 & 0 \times 0 + 2 \times (-1) + 1 \times 1 \3 \times 1 + 2 \times 2 + (-1) \times 3 & 3 \times 0 + 2 \times (-1) + (-1) \times 1\end{array}\right]\]Simplifying each element, we get:\[AB = \left[\begin{array}{cc}2 & -2 \7 & -1 \7 & -3\end{array}\right]\]

Step 4: Check the possibility of finding BA

For \(BA\), check if the multiplication is possible by ensuring the number of columns in \(B\) matches the number of rows in \(A\). Matrix \(B\) has 2 columns, and \(A\) has 3 rows, so the multiplication \(BA\) is not possible.

Key Concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is crucial. The dimensions of a matrix are denoted by the number of rows and columns it contains. This is written in the form \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
For example:

• Matrix \( A \) in the problem is a \( 3 \times 3 \) matrix, indicating it has 3 rows and 3 columns.
• Matrix \( B \) is a \( 3 \times 2 \) matrix, meaning it has 3 rows and 2 columns.

Knowing the dimensions is the first step in determining the possibility of matrix multiplication. The dimensions tell us how to align the matrices to multiply them correctly.

Matrix Multiplication Rules

Matrix multiplication is not as straightforward as regular number multiplication. There are specific rules you need to follow:
One of the most important rules for multiplying two matrices is that the number of columns in the first matrix must match the number of rows in the second matrix.
For example:

• To multiply \( A \) and \( B \) (i.e., find \( AB \)), check that matrix \( A \) has 3 columns, and matrix \( B \) has 3 rows. As these numbers match, multiplication is possible.

When verifying if \( BA \) is possible, notice that the transition is not viable because matrix \( B \) has only 2 columns while matrix \( A \) demands 3 rows, which breaks the rule, meaning \( BA \) cannot be computed.
Another note is that even if both multiplications \( AB \) and \( BA \) were possible, they could yield different results, as matrix multiplication is not commutative. Always check the dimensions!

Matrix Product Calculation

Once verifying dimensions and ensuring rules are followed, you can calculate the matrix product.
The process involves taking each row of the first matrix and multiplying it by every column of the second matrix. Here's how you do it:

• Start with the first row of Matrix \( A \) and multiply it by each column of Matrix \( B \), summing the products.
• Continue this process for each row in \( A \).
• For example, the element in the first row, first column of \( AB \) is computed as: \( 2 \times 1 + 1 \times 2 + (-1) \times 3 = 2 \).

This calculation must be repeated for each element in the resulting matrix.
The resulting product matrix \( AB \) from our exercise is:

• First row: \( \left[2, -2\right] \)
• Second row: \( \left[7, -1\right] \)
• Third row: \( \left[7, -3\right] \)

Understanding the way to multiply matrices ensures accuracy and comprehension when dealing with more complex problems.