Question

If a matrix X is such that \(\begin{bmatrix}2 & 1\end{bmatrix}X=\begin{bmatrix}3 & 4 & 5\end{bmatrix}\), then the order of matrix X is :

• A. \(3\times 1\)
• B. \(2\times 3\)
• C. \(1\times 2\)
• D. \(1\times 3\)

Hint:

Columns of the first matrix must equal rows of X. The columns of the result equal the columns of X.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The product \(AB\) exists only when the number of columns of \(A\) equals the number of rows of \(B\).
If \(A\) is \(m\times n\) and \(B\) is \(n\times p\), the product has order \(m\times p\).

Step 2: Key Formula or Approach:
Write down the orders of the known matrices. Then let X have order \(r\times s\) and match the rules.

Step 3: Detailed Explanation:
The matrix \(\begin{bmatrix}2 & 1\end{bmatrix}\) has 1 row and 2 columns, so its order is \(1\times 2\).
The right side \(\begin{bmatrix}3 & 4 & 5\end{bmatrix}\) has order \(1\times 3\).
Let X have order \(r\times s\).
The columns of the left matrix (2) must equal the rows of X, so \(r=2\).
The product has order \(1\times s\). It must equal \(1\times 3\), so \(s=3\).
So X has order \(2\times 3\).

Step 4: Why the other options are wrong:
Option (A) \(3\times 1\) has 3 rows, but we need 2 rows, so the product cannot be formed.
Option (C) \(1\times 2\) has 1 row, so again the product is not defined.
Option (D) \(1\times 3\) has 1 row, which fails in the same way.

Final Answer:
The order of X is \(2\times 3\), option (B). \[ \boxed{2\times 3} \]