Problem 41 - A Graphical Approach to Precalculus with Limits

Question

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 4 \times 2 ; B \text { is } 2 \times 4$$

Step-by-Step Solution

Answer

Dimensions of AB: \(4 \times 4\). BA is not defined.

Step 1: Understand Matrix Multiplication Requirements

For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, matrix \(A\) is \(4 \times 2\) and matrix \(B\) is \(2 \times 4\). Therefore, the product \(AB\) is defined because the number of columns in \(A\) (2) matches the number of rows in \(B\) (2). Similarly, check \(BA\): the number of columns in \(B\) (4) does not match the number of rows in \(A\) (4), so \(BA\) is not defined.

Step 2: Calculate Dimensions of Product AB

Once we know that the product \(AB\) is defined, the dimensions of the resulting matrix is determined by the number of rows of the first matrix and the number of columns of the second matrix. Therefore, the dimensions of \(AB\) will be \(4 \times 4\).

Key Concepts

Matrix Dimensions

When talking about matrices, dimensions are a crucial concept. The dimensions of a matrix are indicated by the number of rows and columns it contains. Think of a matrix as a table, where each element is stored in a particular row and column. The dimensions are expressed in terms of rows and then columns, like this: \(m \times n\). Here, \(m\) is the number of rows, and \(n\) is the number of columns.
For example, if you have a matrix \(A\) with dimensions \(4 \times 2\), it means that there are 4 rows and 2 columns in \(A\). Knowing the dimensions is the first step in understanding how matrices interact with each other, especially in operations like multiplication. It's essential to get comfortable with matrix dimensions because they dictate what kinds of operations are possible.

Matrix Product

Multiplying two matrices together requires following specific rules, one of which is called the multiplication condition: the number of columns in the first matrix must be the same as the number of rows in the second matrix. If this requirement is met, the product of the two matrices is defined.
This product is a new matrix whose dimensions are determined by the number of rows from the first matrix and the number of columns from the second matrix.

• For the matrices \(A\) and \(B\), where \(A\) is \(4 \times 2\) and \(B\) is \(2 \times 4\), the product \(AB\) can be computed because the number of columns in \(A\) (which is 2) matches the number of rows in \(B\) (which is also 2).
• The resulting matrix \(AB\) will therefore have dimensions \(4 \times 4\).

This outcome results from matching each row of the first matrix with each column of the second matrix, essentially conducting a combination of dot products to build a new matrix. Remember that if the condition is not met, like trying to calculate \(BA\), the product is not defined.

Dimensional Analysis

Dimensional analysis in the context of matrices refers to the careful consideration of the dimensions when performing operations like multiplication. This discipline ensures that operations like matrix multiplication are valid and meaningful.
Not every pair of matrices can be multiplied—it's only possible when the number of columns in the first matrix equals the number of rows in the second matrix.
If a valid multiplication is possible, the resulting matrix's dimensions are dictated by the rows of the first matrix and the columns of the second matrix.

• For example, multiplying a \(4 \times 2\) matrix with a \(2 \times 4\) matrix results in a \(4 \times 4\) matrix.
• Conversely, switching the order to attempt a \(B \times A\) product, which requires matching \(4\) columns to \(4\) rows, fails the dimensional analysis, resulting in an undefined product.

Following these dimensional guidelines is key to conducting matrix operations correctly, ensuring accurate mathematical modeling and analysis in more advanced applications.