Problem 73 - Algebra and Trigonometry

Question

Describe when the multiplication of two matrices is not defined.

Step-by-Step Solution

Answer

The multiplication of two matrices is not defined if the number of columns in the first matrix is not equal to the number of rows in the second matrix. For example, if Matrix A has dimensions \(m \times n\) and Matrix B has dimensions \(p \times q\), then \(A \times B\) is not defined if \(n \neq p\).

Step 1: Understanding Matrix Multiplication

Matrix multiplication is not the same as regular multiplication. It requires adherence to specific rules. One important rule is that you can only multiply two matrices if the number of columns in the first matrix is equal to the number of rows in the second matrix.

Step 2: Applying the Rule for Not Defined Multiplication

If the number of columns in the first matrix is not equal to the number of rows in the second matrix, the multiplication of these matrices is not defined. In mathematical terms, for two matrices \(A\) and \(B\), where \(A\) has dimensions of \(m \times n\) and \(B\) has dimensions of \(p \times q\), the multiplication \(A \times B\) will not be defined if \(n \neq p\).

Step 3: Example

Using this rule, consider the two matrices \(X = \left[ \begin{array}{c} 2 \ 5 \ 7 \end{array} \right]\) having dimension \(3 \times 1\) and \(Y = \left[ \begin{array}{ccc} 1 & 2 & 3 \end{array} \right]\) having dimension \(1 \times 3\). Here, the number of columns in \(X\) does not equal to the number of rows in \(Y\). Thus, \(X \times Y\) is not defined.