Matrix Algebra

We are given a real matrix such that:

where and are the identity and null matrices, respectively. We are to find if

Concept Used:

This problem involves expressing higher powers of a matrix in terms of lower powers using polynomial relations (similar to the Cayley-Hamilton theorem).

The given condition is a polynomial identity satisfied by . We can rearrange it to express in terms of and , then use successive multiplication to reach .

Step-by-Step Solution:

Step 1: Expand the given equation:

Step 2: Compute .

Now substitute :

Step 3: Compute .

Substitute :

Hence, comparing with :

Final Computation & Result:

Answer: 12

Given:


Step 1: Find such that
First, find .
Now, Multiplying:

Step 2: Find such that
Multiplying:

Step 3: Compute


Step 4: Compute


Step 5: Sum of diagonal elements


Final Answer:

Step 1: Understanding the Concept:
We can express the unknown vector as a linear combination of vectors for which the outputs of are known. By linearity:
Step 2: Key Formula or Approach:
Let Assume: Thus:
Step 3: Detailed Explanation:
Applying matrix : This gives equations: From first: Substitute: Subtract: Then: Thus:
Step 4: Final Answer:
The sum .