If A and B are matrices of order`3 xx n and m xx 5` respectively, then find the order of matrix 5A – 3B, given that it is defined.

Answer

To find the order of the matrix \(5A - 3B\), we first need to understand the orders of the matrices \(A\) and \(B\).

1. **Identify the order of matrices A and B:**
- Matrix \(A\) is given to be of order \(3 \times n\).
- Matrix \(B\) is given to be of order \(m \times 5\).

2. **Condition for matrix addition/subtraction:**
- For the operation \(5A - 3B\) to be defined, both matrices must be of the same order. This means that the number of rows and columns in both matrices must be equal.

3. **Set the orders equal:**
- Since \(A\) is \(3 \times n\) and \(B\) is \(m \times 5\), we can set the dimensions equal:
- The number of rows: \(3 = m\)
- The number of columns: \(n = 5\)

4. **Determine the values of m and n:**
- From the equation \(3 = m\), we find that \(m = 3\).
- From the equation \(n = 5\), we find that \(n = 5\).

5. **Determine the order of the resulting matrix:**
- Since both matrices \(5A\) and \(3B\) are now of the same order, we can conclude that the order of the matrix \(5A - 3B\) will be the same as the order of \(A\) (or \(B\)).
- Therefore, the order of \(5A - 3B\) is \(3 \times 5\).

**Final Answer:**
The order of the matrix \(5A - 3B\) is \(3 \times 5\).

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