Question

Let $X$ be a matrix of order $2 \times n$ and $Z$ be a matrix of order $2 \times p$. If $n = p$, then the order of the matrix $7X - 5Z$ is:

• A. $2 \times n$
• B. $n \times 3$
• C. $p \times 2$
• D. $p \times n$

Hint:

Remember that simple linear combinations of matrices (like $aA + bB$) never change the dimensions. The result is just another matrix taking up the exact same "shape" as the inputs.

Answer 1

The Correct Option is A

Step 1: Identify given orders
\[ X \text{ is of order } 2 \times n, Z \text{ is of order } 2 \times p \] Given $n = p$, so: \[ Z \text{ is also of order } 2 \times n \]
Step 2: Scalar multiplication
Multiplying a matrix by a scalar does not change its order: \[ 7X \text{ has order } 2 \times n, 5Z \text{ has order } 2 \times n \]
Step 3: Subtraction rule
Two matrices can be subtracted only if their orders are the same, and the result has the same order.
Step 4: Resulting order
\[ 7X - 5Z \text{ has order } 2 \times n \] Final Answer:
\[ \boxed{2 \times n} \]