Question

Assertion (A): If \(P = \begin{bmatrix}1 & 4 \\ 2 & 3 \end{bmatrix}\), then \(P^2 - 4P - 5I = 0\).
Reason (R): Every square matrix satisfies its own characteristic equation.

• A. Both (A) and (R) are correct and (R) is correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

To verify matrix equations, always use Cayley-Hamilton theorem—it simplifies calculations drastically.

Answer 1

The Correct Option is A

Concept: Cayley-Hamilton Theorem states: \[ \text{Every square matrix satisfies its own characteristic equation} \]

Step 1: Find characteristic equation of matrix \(P\).
\[ P = \begin{bmatrix}1 & 4 2 & 3 \end{bmatrix} \] Characteristic equation: \[ |P - \lambda I| = 0 \] \[ \begin{vmatrix} 1-\lambda & 4 \\ 2 & 3-\lambda \end{vmatrix} = 0 \]

Step 2: Expand determinant.
\[ (1-\lambda)(3-\lambda) - 8 = 0 \] \[ = (3 - \lambda - 3\lambda + \lambda^2) - 8 \] \[ = \lambda^2 -4\lambda +3 - 8 \] \[ = \lambda^2 -4\lambda -5 = 0 \]

Step 3: Apply Cayley-Hamilton theorem.
Replace \(\lambda\) with matrix \(P\): \[ P^2 - 4P - 5I = 0 \]

Step 4: Verify assertion.
Thus Assertion (A) is correct.

Step 5: Analyze reason.
Reason (R) is exactly Cayley-Hamilton theorem, which explains the result. Final Answer: \[ \boxed{\text{Both (A) and (R) are correct and (R) explains (A)}} \]