If A is a 2*3 Matrix and AB is a 2*5 matrix, then B must be a :

Options

A. 3*5 matrix
B. 5*3 matrix
C. 3*2 matrix
D. 5*2 matrix

Answer

To solve the problem, we need to determine the dimensions of matrix B given the dimensions of matrix A and the product AB.

### Step-by-Step Solution:

1. **Identify the dimensions of matrix A**:
- Matrix A is given as a 2x3 matrix. This means it has 2 rows and 3 columns.

2. **Identify the dimensions of the product AB**:
- The product AB is given as a 2x5 matrix. This means the resulting matrix from the multiplication has 2 rows and 5 columns.

3. **Understand the multiplication rule of matrices**:
- When multiplying two matrices, the number of columns in the first matrix (A) must equal the number of rows in the second matrix (B).
- Therefore, if A is a 2x3 matrix, and we denote the dimensions of matrix B as m x n, we have:
- The number of columns in A = 3 (from A's dimensions)
- The number of rows in B = m (which we need to find)

4. **Set up the equation based on the multiplication rule**:
- From the multiplication rule, we know that:
- The number of columns in A (3) must equal the number of rows in B (m).
- Therefore, we can conclude that:
- m = 3

5. **Determine the dimensions of matrix B**:
- The resulting matrix AB has dimensions 2x5.
- The number of rows in the product (AB) comes from the first matrix (A), which is 2.
- The number of columns in the product (AB) comes from the second matrix (B), which is n.
- Therefore, we can conclude that:
- n = 5

6. **Final dimensions of matrix B**:
- We have found that m = 3 and n = 5.
- Thus, the dimensions of matrix B are 3x5.

### Conclusion:
The order of matrix B must be 3x5.