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

Step-by-Step Solution

Answer

Both products AB and BA are not defined.

Step 1: Check Dimensions for AB

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \( A \) is \( 1 \times 6 \) and matrix \( B \) is \( 2 \times 4 \). Here, the number of columns in \( A \) (which is 6) does not match the number of rows in \( B \) (which is 2). Therefore, the product \( AB \) is not defined.

Step 2: Check Dimensions for BA

Similarly, for the product \( BA \), the number of columns in matrix \( B \) must match the number of rows in matrix \( A \). Matrix \( B \) is \( 2 \times 4 \) and matrix \( A \) is \( 1 \times 6 \). The number of columns in \( B \) (which is 4) does not equal the number of rows in \( A \) (which is 1). Therefore, the product \( BA \) is not defined as well.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for performing matrix operations. Dimensions are described by the number of rows and columns in a matrix. When you see something like "1 × 6", it means the matrix has 1 row and 6 columns. It follows the format of "rows x columns". Knowing dimensions allows us to determine if matrices can be multiplied. It also helps find the size of resulting matrices when products are defined. Always check dimensions before attempting multiplication.

Product of Matrices

In matrix multiplication, not all pairs of matrices can be multiplied. For two matrices, let's say matrix A and matrix B, multiplication is only possible when the number of columns in matrix A equals the number of rows in matrix B. - Example: If matrix A is of size "3 × 2", it can only multiply with a matrix B having size "2 × n". The resulting matrix will then be of size "3 × n".
Matrix multiplication is not like regular multiplication; the order of matrices matters a lot. If matrix A times B is defined, it does not mean B times A is automatically defined. - Remember: Order and dimensions are key!

Undefined Matrix Product

An undefined matrix product occurs when the necessary condition for multiplication is not met. As already mentioned, this condition is that the number of columns in the first matrix should match the number of rows in the second. - For example, consider matrices with dimensions: matrix A (1 × 6) and matrix B (2 × 4). Here, matrix A has more columns than the number of rows in matrix B, so the product AB is not defined. - Similarly, trying to multiply matrix B (2 × 4) by matrix A (1 × 6) fails because the number of columns in B (which is 4) does not match the number of rows in A (which is 1). As a result, the product BA is also undefined.
Always verify dimensions before multiplying to avoid errors. An undefined product simply means multiplication cannot proceed under the given circumstances.