Problem 20 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A B\) and \(B A\). $$A=\left[\begin{array}{lll} 4 & 8 \end{array}\right], \quad B=\left[\begin{array}{r} -3 \\ 2 \end{array}\right]$$

Step-by-Step Solution

Answer

\(AB = \begin{bmatrix} 4 \end{bmatrix}\); \(BA\) is not defined.

Step 1: Check Dimensions for Matrix Multiplication

First, we need to ensure that the matrices can be multiplied. Matrix \(A\) is a \(1 \times 2\) matrix, and matrix \(B\) is a \(2 \times 1\) matrix. Since the number of columns in \(A\) matches the number of rows in \(B\), the multiplication \(AB\) is possible, and it will result in a \(1 \times 1\) matrix.

Step 2: Calculate AB

We multiply matrix \(A\) with matrix \(B\) using the formula for the matrix product: \[AB = \begin{bmatrix} 4 & 8 \end{bmatrix} \begin{bmatrix} -3 \ 2 \end{bmatrix} = (4)(-3) + (8)(2) = -12 + 16 = 4\]Thus, the product \(AB\) is \(\begin{bmatrix} 4 \end{bmatrix}\).

Step 3: Check Dimensions for Multiplication BA

Next, check if \(BA\) is possible. Matrix \(B\) is a \(2 \times 1\) matrix and \(A\) is a \(1 \times 2\) matrix. Since the number of columns in \(B\) does not match the number of rows in \(A\), multiplication \(BA\) is not defined.

Key Concepts

Understanding Matrix Dimensions

Matrix dimensions are fundamental to determine whether two matrices can be multiplied. Each matrix has dimensions expressed as "rows × columns," indicating how many rows and columns the matrix contains. For instance, matrix \(A\) in the exercise is a \(1 \times 2\) matrix, which means it has 1 row and 2 columns. Meanwhile, matrix \(B\) is a \(2 \times 1\) matrix, representing 2 rows and 1 column.

The dimensions are crucial since they dictate the possibility of multiplication between two matrices. In basic terms, you need to ensure that the number of columns in the first matrix matches the number of rows in the second matrix.

• If the inner dimensions match (columns of the first matrix and rows of the second), multiplication can proceed.
• The resulting matrix will then have dimensions determined by the outer dimensions.

Calculating the Matrix Product

Matrix multiplication involves calculating the matrix product by performing a sequence of operations between the elements of the matrix. In the given exercise, you successfully calculated the product of matrices \(A\) and \(B\) to get \(AB\). When we multiply a \(1 \times 2\) matrix (like \(A\)) by a \(2 \times 1\) matrix (like \(B\)), we can proceed by:

• Taking each element from the row of the first matrix and the corresponding element from the column of the second matrix.
• Calculating the sum of their products to get a single number.

The result for \(AB\) is a new matrix of dimension \(1 \times 1\). In this case, you calculated:
\[AB = \begin{bmatrix} 4 & 8 \end{bmatrix} \begin{bmatrix} -3 \ 2 \end{bmatrix} = (4 \times -3) + (8 \times 2) = -12 + 16 = 4\]
The outcome \(\begin{bmatrix} 4 \end{bmatrix}\) is a scalar in a \(1 \times 1\) matrix.

Matrix Compatibility in Multiplication

Matrix compatibility is essential to determine if two matrices can be multiplied. This concept revolves around the requirement that the number of columns of the first matrix must equal the number of rows of the second matrix. In our exercise:

• For \(AB\), matrix \(A\) has 2 columns, which matches the 2 rows in matrix \(B\), making their multiplication possible.
• On the other hand, for \(BA\), matrix \(B\) has 1 column, while matrix \(A\) has 1 row. While these dimensions are equal, \(BA\)'s situation shifts the order, resulting in incompatibility.

Understanding compatibility ensures efficient computation and prevents errors during problem-solving. If matrices are not compatible, you cannot find a product, as matrices operations depend on their structural harmony.