If A is a matrix of order `[x- 1] xx 3` and B is matrix of order `3 xx [y-2]`,where [ ] represent greatest integer function, such that AB is a matrix of order `4xx5` then

Options

A. `x in[5,6)`
B. `x in[5,6]`
C. `yin[7,8]`
D. `yin[8,9]`

Answer

To solve the problem step by step, we need to analyze the given information about the matrices A and B and their product AB.

### Step 1: Understand the orders of matrices A and B
- Matrix A has an order of \([x - 1] \times 3\).
- Matrix B has an order of \(3 \times [y - 2]\).

### Step 2: Determine the order of the product AB
- The product AB will have an order of \([x - 1] \times [y - 2]\).
- We are given that the order of AB is \(4 \times 5\).

### Step 3: Set up the equations based on the orders
From the order of the product AB, we can write:
\[ [x - 1] = 4 \quad \text{and} \quad [y - 2] = 5 \]

### Step 4: Solve for x
From \([x - 1] = 4\):
\[ 4 \leq x - 1 < 5 \]
Adding 1 to all parts of the inequality:
\[ 5 \leq x < 6 \]
Thus, \(x\) can take values in the interval:
\[ x \in [5, 6) \]

### Step 5: Solve for y
From \([y - 2] = 5\):
\[ 5 \leq y - 2 < 6 \]
Adding 2 to all parts of the inequality:
\[ 7 \leq y < 8 \]
Thus, \(y\) can take values in the interval:
\[ y \in [7, 8) \]

### Conclusion
The values of \(x\) and \(y\) are:
- \(x \in [5, 6)\)
- \(y \in [7, 8)\)