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Chapter 9: Problem 74

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

Short Answer

Expert verified
The order or dimensions of the product of two matrices A and B is determined by the number of rows of the first matrix and the number of columns of the second matrix, that is, if A is \(p \times q\) and B is \(r \times s\), and if \(q = r\), then the product matrix is \(p \times s\).

Step by step solution

01

Understand the rule of matrix multiplication

Matrix multiplication is possible only when the number of columns of the first matrix equals the number of rows of the second matrix. The key point to remember is if matrix A has dimensions \(p \times q\) and matrix B has dimensions \(r \times s\), then A and B can be multiplied (in that order) only if \(q = r\). This rule ensures that each element in a row of the first matrix corresponds to an element in a column of the second matrix.
02

Determine the dimensions of the product

The product of the matrix multiplication will form a new matrix. The order or dimensions of this resulting matrix will have the same number of rows as the first matrix (p from matrix A) and the same number of columns as the second matrix (s from matrix B). So, the resulting matrix from a multiplication of matrix A (\(p \times q\)) and matrix B (\(r \times s\)) will have dimensions \(p \times s\). In other words, the outer dimensions from the original matrices determine the dimensions of the product matrix.
03

Put the rules into practice

To illustrate these rules, consider a practical example where Matrix A is a \(2 \times 3\) matrix and Matrix B is a \(3 \times 4\) matrix. According to the rules, these matrices can be multiplied since the number of columns in A (3) equals the number of rows in B (3). The product matrix will have dimensions defined by taking the outer values, which in this case are 2 (number of rows of matrix A) and 4 (number of columns of matrix B). Thus, the resulting matrix will have dimensions \(2 \times 4\).

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