Problem 45 - A Graphical Approach to College Algebra

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. $$A \text { is } 4 \times 3 ; B \text { is } 2 \times 5.$$

Step-by-Step Solution

Answer

Both products \(AB\) and \(BA\) are undefined.

Step 1: Check the Definition of Matrix Multiplication

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. We will apply this rule to matrices \(A\) and \(B\).

Step 2: Evaluate the Product \(AB\)

Matrix \(A\) has dimensions \(4 \times 3\), and matrix \(B\) has dimensions \(2 \times 5\). Compare the number of columns in \(A\) (which is 3) with the number of rows in \(B\) (which is 2). Since these numbers are different, the product \(AB\) is undefined.

Step 3: Evaluate the Product \(BA\)

Matrix \(B\) has dimensions \(2 \times 5\), and matrix \(A\) has dimensions \(4 \times 3\). Compare the number of columns in \(B\) (which is 5) with the number of rows in \(A\) (which is 4). These numbers are not equal, so the product \(BA\) is also undefined.

Key Concepts

Dimensions of Matrices

Matrices play a crucial role in linear algebra, and understanding their dimensions is the first step in mastering matrix operations. The dimensions of a matrix are defined by the number of rows and columns it contains. For example, if we say that matrix \( A \) is \( 4 \times 3 \), it means that matrix \( A \) has 4 rows and 3 columns. Similarly, if matrix \( B \) is \( 2 \times 5 \), it indicates that matrix \( B \) has 2 rows and 5 columns.
Understanding the dimensions helps in performing matrix operations like addition, subtraction, and multiplication. It is important because these dimensions determine whether certain operations, such as multiplication, can be performed.
To summarize, the dimensions of a matrix \( M \times N \) imply that:

• \( M \) is the number of rows
• \( N \) is the number of columns

Matrix Multiplication Rules

Matrix multiplication is a fundamental operation where we take two matrices and produce a third matrix. However, not all matrices can be multiplied. To multiply two matrices, you must follow a key rule: the number of columns in the first matrix must equal the number of rows in the second matrix.
For example, consider matrices \( A \) and \( B \) with dimensions \( 4 \times 3 \) and \( 2 \times 5 \), respectively. Here’s how you apply the rule:

• Check if the number of columns in \( A \) (which is 3) matches the number of rows in \( B \) (which is 2). They do not match, so \( AB \) is not defined.
• Now, check if the number of columns in \( B \) (which is 5) matches the number of rows in \( A \) (which is 4). They do not match, so \( BA \) is not defined.

In matrix multiplication, following this rule guarantees that the operation can proceed, and the resulting product will have a predictable dimension of \( M \times N \) where \( M \) is from the first matrix and \( N \) is from the second.

Undefined Matrix Product

The concept of an undefined matrix product arises when the rule for matrix multiplication is not met. When trying to multiply matrix \( A \) by matrix \( B \), it is crucial to compare the columns of \( A \) to the rows of \( B \). If these numbers do not match, the product is undefined. The same logic applies when reversing the order to multiply matrix \( B \) by matrix \( A \).
In our given example, matrix \( AB \) is undefined because \( A \) has 3 columns and \( B \) has 2 rows. Similarly, the product \( BA \) is also undefined as \( B \) has 5 columns and \( A \) has 4 rows. In both cases, the necessary condition for multiplication isn't satisfied.
It's important to verify dimensions before attempting multiplication, to avoid undefined products. Doing so will ensure that you are only working with well-formed and computable matrices ensuring your work in linear algebra is accurate and meaningful.