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

To determine the order of the matrix expression \(7X - 5Z\), we need to understand the orders of the matrices \(X\) and \(Z\), and how matrix operations affect them.

1. Given:
   • \(X\) is a matrix of order \(2 \times n\).
   • \(Z\) is a matrix of order \(2 \times p\).
   • It is provided that \(n = p\).
2. Since \(n = p\), both matrices \(X\) and \(Z\) have the same number of columns, i.e., \(n\).
3. The expression \(7X - 5Z\) indicates a linear combination of matrices \(X\) and \(Z\). This operation is possible only if the matrices have the same dimensions.
4. As both matrices \(X\) and \(Z\) have dimensions \(2 \times n\), the operation \(7X - 5Z\) is valid and results in a matrix of the same dimensions as \(X\) and \(Z\).
5. Therefore, the order of the matrix \(7X - 5Z\) is \(2 \times n\).

Conclusion: The correct answer is \(2 \times n\).