Problem 52 - A Graphical Approach to Precalculus with Limits

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

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

Step 1: Determine the Compatibility of Multiplication

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Matrix \(A\) is a \(3 \times 3\) matrix, and matrix \(B\) is a \(3 \times 2\) matrix. Since the number of columns in \(A\) (3) matches the number of rows in \(B\) (3), \(A \times B\) is possible.

Step 2: Calculate Product \(AB\)

To find \(AB\), we multiply each row of \(A\) by each column of \(B\) and sum the products:- First row, first column: \((2)(1) + (1)(2) + (-1)(3) = 2 + 2 - 3 = 1\)- First row, second column: \((2)(0) + (1)(-1) + (-1)(1) = 0 - 1 - 1 = -2\)- Second row, first column: \((0)(1) + (2)(2) + (1)(3) = 0 + 4 + 3 = 7\)- Second row, second column: \((0)(0) + (2)(-1) + (1)(1) = 0 - 2 + 1 = -1\)- Third row, first column: \((3)(1) + (2)(2) + (-1)(3) = 3 + 4 - 3 = 4\)- Third row, second column: \((3)(0) + (2)(-1) + (-1)(1) = 0 - 2 - 1 = -3\)The product \(AB\) is:\[AB = \begin{bmatrix}1 & -2 \7 & -1 \4 & -3\end{bmatrix}\]

Step 3: Determine Compatibility of Reverse Multiplication

Now, consider \(B A\). Matrix \(B\) is a \(3 \times 2\) matrix and \(A\) is a \(3 \times 3\) matrix. For \(B A\) to be defined, the number of columns in \(B\) (2) must match the number of rows in \(A\) (3). Since they do not match, \(B A\) is not possible.

Key Concepts

Matrix Compatibility

Before we dive into multiplying matrices, it's crucial to check if they are compatible for multiplication. This is all about matrix dimensions. If you're dealing with two matrices, say Matrix A and Matrix B, their dimensions must meet a specific criterion.
Matrix A's number of columns must be the same as Matrix B's number of rows to find their product. This compatibility check ensures that each element in a row of Matrix A can be paired with an element in a column of Matrix B.

• Example: If A is a 3x3 matrix, and B is a 3x2 matrix, A's 3 columns can partner with B's 3 rows.
• This compatibility means you can multiply A and B to get a new matrix.

However, when checking B × A, B's 2 columns don't match A's 3 rows, making this multiplication impossible.

Matrix Product

Once matrices are compatible, you can move on to finding the matrix product. This involves a bit of methodical multiplication and addition. Let's take Matrix A and B as examples.
When computing the matrix product AB, you go through every row of A and column of B. Multiply corresponding elements and sum the results to form each element of the resulting matrix.

• Step-by-step:
• Multiply each element of the first row of A by the elements of the first column of B.
• Add these up to get the element in the first column of your product matrix.
• Repeat for each row of A and each column of B to fill out the entire product matrix.

In our example:

• First row, first column: (2)(1) + (1)(2) + (-1)(3) = 1
• The final matrix product AB is:\[AB = \begin{bmatrix}1 & -2 \ 7 & -1 \ 4 & -3\end{bmatrix}\]

Matrix Dimensions

Understanding matrix dimensions is foundational when working with matrices in algebra. Matrix dimensions are described by two numbers: the number of rows followed by the number of columns.
For example, a matrix with 3 rows and 3 columns is a 3x3 matrix. Dimensions are key when performing operations like addition, subtraction, or multiplication with matrices.

• Significance: Dimensions guide you in understanding matrix compatibility. For multiplication to occur, inner dimensions (columns of the first matrix and rows of the second one) must match.
• Result: The dimensions of the resulting matrix from multiplication are determined by the outer numbers—the number of rows from the first matrix and columns from the second matrix.
• In the case of A (3x3) and B (3x2), multiplying gives a 3x2 matrix.