Determinant

Step 1: Property of determinants
For a square matrix , , where is the size of .

Step 2: Compute
Using cofactor expansion:
Step 3: Calculate
Since :
Step 4: Verify the options
The correct value is , which corresponds to option (D).

Given that , we need to find matrix .

The inverse of a matrix is given by: Applying this to : The determinant of is: Now, the inverse of (which is ) is: Therefore, the correct answer is:

Rewriting the equation: This is a first-order linear differential equation of the form:
Final Answer:

The determinant is:

Simplify:

Given , solve:

Final Answer: