Problem 46 - 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. $$A \text { is } 1 \times 6 ; B \text { is } 2 \times 4$$

Step-by-Step Solution

Answer

Neither \(AB\) nor \(BA\) is defined.

Step 1: Understanding Matrix Multiplication

Matrix multiplication is defined when the number of columns in the first matrix equals the number of rows in the second matrix. The resulting product has dimensions equal to the number of rows in the first matrix and the number of columns in the second matrix.

Step 2: Checking Multiplicability of AB

Matrix \(A\) is \(1 \times 6\) and matrix \(B\) is \(2 \times 4\). For the multiplication \(AB\) to be defined, the number of columns in \(A\) (6) must equal the number of rows in \(B\) (2). Since they do not match, the product \(AB\) is not defined.

Step 3: Checking Multiplicability of BA

Matrix \(B\) is \(2 \times 4\) and matrix \(A\) is \(1 \times 6\). For the multiplication \(BA\) to be defined, the number of columns in \(B\) (4) must equal the number of rows in \(A\) (1). Since they do not match, the product \(BA\) is not defined.

Key Concepts

Matrix Dimensions

When dealing with matrices, it's essential to understand the concept of their dimensions. A matrix is essentially an array of numbers arranged in rows and columns. The size or dimensions of a matrix are described in terms of its number of rows and columns.
For example, if a matrix has 1 row and 6 columns, we call it a \(1 \times 6\) matrix. Similarly, a \(2 \times 4\) matrix includes 2 rows and 4 columns.

• The first number in the dimension indicates the number of rows.
• The second number indicates the number of columns.

Understanding dimensions is crucial as it directly affects whether you can multiply matrices together. Whenever you encounter a matrix, always start by identifying its dimensions as this provides the groundwork for further operations.

Multiplicability

Matrix multiplicability depends strictly on the dimensions of the matrices involved. To determine if two matrices can be multiplied, you should compare the dimensions. Matrix multiplication is permissible only if the number of columns in the first matrix is the same as the number of rows in the second matrix.
For instance, to multiply matrix \(A\) with dimensions \(1 \times 6\) by matrix \(B\) with dimensions \(2 \times 4\), you check the match between the number of columns in \(A\) (which is 6) and the number of rows in \(B\) (which is 2). Since 6 does not equal 2, matrices \(A\) and \(B\) cannot be multiplied in the order \(AB\).

• The multiplication \(AB\) is not defined since the necessary condition \(\text{columns of } A = \text{rows of } B\) is not satisfied.
• This rule helps prevent errors when working with matrix operations and ensures that the multiplication process is meaningful and correct.

Undefined Product

An undefined matrix product occurs when the matrices involved do not satisfy the necessary condition for multiplication, specifically the matched dimensions rule.
In our example, both matrix products, \(AB\) and \(BA\), are undefined because their dimensions do not allow for multiplication.

• For \(AB\) to be defined, the number of columns in \(A\)\( (6) \) would need to match the number of rows in \(B\)\( (2) \), which they do not.
• For \(BA\) to be defined, the number of columns in \(B\)\( (4) \) would have to equal the number of rows in \(A\)\( (1) \), which again, they do not.

Whenever you check for matrix multiplication potential, always verify these dimensional conditions first. If they aren't met, the product is undefined, meaning you cannot perform the multiplication operation. By understanding this concept, students can avoid confusion and correctly apply matrix operations.