Question

Let $B$ be a matrix of order $3 \times 2$ and $C$ be a matrix of order $3 \times 3$. If $A$ is a matrix such that $BA = C$, then the order of $A$ is

• A. $2 \times 2$
• B. $2 \times 3$
• C. $3 \times 2$
• D. $3 \times 4$
• E. $3 \times 3$

Hint:

$(m \times n) \times (n \times p) \to (m \times p)$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question involves the rules of matrix multiplication. For the product of two matrices to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
Step 2: Key Formula or Approach:
If a matrix of order \( m \times n \) is multiplied by a matrix of order \( p \times q \), the multiplication is possible only if \( n = p \). The resulting matrix will have the order \( m \times q \).
We are given:
Order of B = \( 3 \times 2 \)
Order of C = \( 3 \times 3 \)
Equation: \( B \cdot A = C \)
Let the order of matrix A be \( p \times q \).
Step 3: Detailed Explanation:
The multiplication is \( B_{3 \times 2} \cdot A_{p \times q} = C_{3 \times 3} \).
Condition for multiplication:
For the product BA to be defined, the number of columns of B must equal the number of rows of A.
Number of columns of B = 2
Number of rows of A = p
Therefore, we must have \( p = 2 \).
So, the order of A is \( 2 \times q \).
Order of the resulting matrix:
The product BA will have the order (number of rows of B) \( \times \) (number of columns of A).
Order of BA = \( 3 \times q \).
We are given that BA = C, and the order of C is \( 3 \times 3 \).
So, the order of BA must be the same as the order of C.
\( 3 \times q = 3 \times 3 \)
By comparing the dimensions, we find that \( q = 3 \).
Combining our findings, the order of matrix A is \( p \times q = 2 \times 3 \).
Step 4: Final Answer:
The order of matrix A is 2 x 3.