Question:
If \(\begin{bmatrix} 4 1 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix}\), then order of \(A\) must be :
If \(\begin{bmatrix} 4 1 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix}\), then order of \(A\) must be :
Show Hint
Always visualize the order as \((3 \times 1) \cdot (1 \times 3) = (3 \times 3)\).
The "inner" numbers (1 and 1) must match for the operation to be valid, and the "outer" numbers (3 and 3) determine the size of the final matrix.
Updated On: Sep 10, 2026
- \(3 \times 1\)
- \(1 \times 3\)
- \(1 \times 1\)
- \(3 \times 3\)
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The Correct Option is B
Solution and Explanation
Concept:
• Multiplication of two matrices is only possible if the number of columns in the first matrix equals the number of rows in the second matrix.
• If matrix \(P\) is of order \(m \times n\) and matrix \(Q\) is of order \(n \times p\), their product \(PQ\) will be of order \(m \times p\).
Step 1: Determine the order of the known matrices
Let the column matrix on the left be \(M\). It has \(3\) rows and \(1\) column, so its order is \(3 \times 1\). The result matrix on the right is a square matrix with \(3\) rows and \(3\) columns, so its order is \(3 \times 3\).
Step 2: Define the order of matrix \(A\)
Let the order of matrix \(A\) be \(r \times c\), where \(r\) is the number of rows and \(c\) is the number of columns.
Step 3: Use the compatibility rule for multiplication
For the product \(M \cdot A\) to be defined: The columns of \(M\) must equal the rows of \(A\). \[ 1 = r \] So, \(A\) must have \(1\) row.
Step 4: Use the rule for the order of the product
The order of the resulting product is (rows of first matrix) \(\times\) (columns of second matrix). The order of the product is \(3 \times c\). Comparing this with the given product order (\(3 \times 3\)): \[ c = 3 \] So, \(A\) must have \(3\) columns.
Step 5: Conclusion
The order of matrix \(A\) is \(r \times c = 1 \times 3\).
• Multiplication of two matrices is only possible if the number of columns in the first matrix equals the number of rows in the second matrix.
• If matrix \(P\) is of order \(m \times n\) and matrix \(Q\) is of order \(n \times p\), their product \(PQ\) will be of order \(m \times p\).
Step 1: Determine the order of the known matrices
Let the column matrix on the left be \(M\). It has \(3\) rows and \(1\) column, so its order is \(3 \times 1\). The result matrix on the right is a square matrix with \(3\) rows and \(3\) columns, so its order is \(3 \times 3\).
Step 2: Define the order of matrix \(A\)
Let the order of matrix \(A\) be \(r \times c\), where \(r\) is the number of rows and \(c\) is the number of columns.
Step 3: Use the compatibility rule for multiplication
For the product \(M \cdot A\) to be defined: The columns of \(M\) must equal the rows of \(A\). \[ 1 = r \] So, \(A\) must have \(1\) row.
Step 4: Use the rule for the order of the product
The order of the resulting product is (rows of first matrix) \(\times\) (columns of second matrix). The order of the product is \(3 \times c\). Comparing this with the given product order (\(3 \times 3\)): \[ c = 3 \] So, \(A\) must have \(3\) columns.
Step 5: Conclusion
The order of matrix \(A\) is \(r \times c = 1 \times 3\).
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