Inverse Of A Matrix

We are asked to differentiate with respect to , so we need to use the chain rule.
Step 2: Verify the options
The correct derivative is , matching option (A).

Step 1: Differentiate using the chain rule
We have:
Step 2: Simplify derivatives
Substitute back: Conclusion:
The derivative is .

Step 1: Rewriting the equation
Divide through by : Step 2: Find the integrating factor
The integrating factor is determined by identifying the dependency on and multiplying the equation by .
Step 3: Verify integrating factor
After multiplying, the left-hand side becomes exact.
The integrating factor is , which matches option (D).

Step 1: Understand the series : The given series is , which is an infinite series. It can be expressed as: if the matrix is invertible.
Step 2: Compute : The identity matrix is: So, .
Step 3: Check if is invertible: The determinant of is: Since the determinant is non-zero, is invertible.
Step 4: Verify the series sum: The inverse of is: Thus, the sum of the series is:
Conclusion: The correct option is .

The given differential equation is: This is a linear first-order differential equation of the form: where and .
Step 1: Find the integrating factor
The integrating factor is given by the formula: Substitute into the equation: The integral of is , so: Since , we can write:
Step 2: Conclusion
Thus, the integrating factor is , which corresponds to option (B).