Problem 61 - College Algebra

Question

If two matrices can be multiplied, describe how to determine the order of the product.

Step-by-Step Solution

Answer

The order of the product of two matrices is determined by the number of rows in the first matrix and the number of columns in the second matrix. Specifically, if the first matrix \( A \) is of order \( m \times n \) and the second matrix \( B \) is of order \( n \times p \), the resulting matrix \( C = AB \) will have an order of \( m \times p \).

Step 1: Understand Matrix Order

The order of a matrix is given as the number of rows by the number of columns. It is typically noted as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.

Step 2: Matrix Multiplication Rule

Two matrices \( A \) and \( B \) can be multiplied if the number of columns in \( A \) is equal to the number of rows in \( B \). If say, matrix \( A \) is of order \( m \times n \) and matrix \( B \) is of order \( n \times p \), the resultant matrix \( C = AB \) will be of the order \( m \times p \). That is, the order of the resulting matrix takes the form of the number of rows of the first matrix by the number of columns of the second matrix.

Step 3: Applying this Rule

Applying this rule will enable us to determine the order of the product when two matrices are multiplied. For instance, if we have two matrices, one of order \( 2 \times 3 \) and the other of order \( 3 \times 4 \), the order of the resultant matrix will be \( 2 \times 4 \).