Question:
If a matrix B is such that \( B \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 0 & 1 & 1 2 & 0 & 1 \end{bmatrix} \), then the order of matrix B is :
If a matrix B is such that \( B \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 0 & 1 & 1 2 & 0 & 1 \end{bmatrix} \), then the order of matrix B is :
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Order matching equation: \( (m \times \underline{n}) \times (\underline{1} \times 3) = (3 \times 3) \). The inside numbers must match (\( n = 1 \)), and the outside numbers must equal the product's dimensions (\( m = 3 \)).
Updated On: Jul 4, 2026
- \( 1 \times 3 \)
- \( 3 \times 1 \)
- \( 3 \times 3 \)
- \( 1 \times 1 \)
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The Correct Option is B
Solution and Explanation
Concept:
The rule for matrix multiplication states that if matrix \( X \) has order \( m \times n \) and matrix \( Y \) has order \( n \times p \), then the resulting matrix \( XY \) is defined and has an order of \( m \times p \).
Step 1: Identify the orders of the given matrices.
Let the unknown order of matrix \( B \) be \( m \times n \). The row matrix given is \( C = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \). It has 1 row and 3 columns, so its order is \( 1 \times 3 \). The resulting product matrix on the right-hand side is a square matrix with 3 rows and 3 columns, so its order is \( 3 \times 3 \).
Step 2: Apply the matrix multiplication compatibility rule.
The product \( B \cdot C \) is defined, which implies that the number of columns in \( B \) must equal the number of rows in \( C \): \[ n = 1 \]
Step 3: Determine the rows using the product order.
The order of the product matrix \( BC \) is given by the number of rows of \( B \) and the number of columns of \( C \), which is \( m \times 3 \). We are given that the product matrix has order \( 3 \times 3 \). Therefore: \[ m = 3 \]
Step 4: Combine the dimensions to find the order of B.
Since \( m = 3 \) and \( n = 1 \), the order of matrix \( B \) is \( 3 \times 1 \). This corresponds exactly to option (B).
Step 1: Identify the orders of the given matrices.
Let the unknown order of matrix \( B \) be \( m \times n \). The row matrix given is \( C = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \). It has 1 row and 3 columns, so its order is \( 1 \times 3 \). The resulting product matrix on the right-hand side is a square matrix with 3 rows and 3 columns, so its order is \( 3 \times 3 \).
Step 2: Apply the matrix multiplication compatibility rule.
The product \( B \cdot C \) is defined, which implies that the number of columns in \( B \) must equal the number of rows in \( C \): \[ n = 1 \]
Step 3: Determine the rows using the product order.
The order of the product matrix \( BC \) is given by the number of rows of \( B \) and the number of columns of \( C \), which is \( m \times 3 \). We are given that the product matrix has order \( 3 \times 3 \). Therefore: \[ m = 3 \]
Step 4: Combine the dimensions to find the order of B.
Since \( m = 3 \) and \( n = 1 \), the order of matrix \( B \) is \( 3 \times 1 \). This corresponds exactly to option (B).
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