Question

If the orders of the matrices A and B are \(m \times n\) and \(n \times p\) respectively, then the order of AB will be

• A. \(m \times p\)
• B. \(p \times m\)
• C. \(m \times n\)
• D. \(n \times p\)

Hint:

A simple way to remember the rule for matrix multiplication order is to write the dimensions side-by-side: \((m \times n) \times (n \times p)\). If the two "inner" numbers are the same (\(n\)), multiplication is possible. The "outer" numbers (\(m\) and \(p\)) give the dimensions of the resulting matrix: \(m \times p\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests the fundamental rule for the multiplication of two matrices. The product of two matrices, AB, is defined only if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B). The resulting matrix has an order determined by the outer dimensions of the original matrices.
Step 2: Key Formula or Approach:
Let matrix A have order \(m \times n\).
Let matrix B have order \(n \times p\).
The condition for multiplication is that the inner dimensions must match: \(n = n\). This is satisfied.
The order of the product matrix AB is given by the outer dimensions: \(m \times p\).
Step 3: Detailed Explanation:
We are given:
Order of matrix A = \(m \times n\) (m rows, n columns).
Order of matrix B = \(n \times p\) (n rows, p columns).
To find the order of the product AB, we look at the dimensions:
\[ (\text{A})_{m \times n} \times (\text{B})_{n \times p} \] The number of columns of A is \(n\), and the number of rows of B is \(n\). Since they are equal, the product AB is defined.
The order of the resulting matrix AB is the number of rows of A by the number of columns of B.
Therefore, the order of AB is \(m \times p\).
Step 4: Final Answer:
The order of the product matrix AB is \(m \times p\).