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.
In the light of the above statements, choose the most appropriate answer from the options given below :
Verifying Assertion (A): Matrix Equation
Given the matrix $P = \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}$, we need to verify the equation $P^{2}-4P-5I=0$.
- Calculate $P^2$: $P^2 = P \times P = \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix} \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}$ $P^2 = \begin{bmatrix}(1 \times 1 + 4 \times 2) & (1 \times 4 + 4 \times 3) \\ (2 \times 1 + 3 \times 2) & (2 \times 4 + 3 \times 3)\end{bmatrix}$ $P^2 = \begin{bmatrix}1 + 8 & 4 + 12 \\ 2 + 6 & 8 + 9\end{bmatrix} = \begin{bmatrix}9 & 16 \\ 8 & 17\end{bmatrix}$
- Calculate $4P$: $4P = 4 \times \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix} = \begin{bmatrix}4 & 16 \\ 8 & 12\end{bmatrix}$
- Calculate $5I$ (where $I$ is the $2 \times 2$ identity matrix): $5I = 5 \times \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}5 & 0 \\ 0 & 5\end{bmatrix}$
- Evaluate $P^{2}-4P-5I$: $P^{2}-4P-5I = \begin{bmatrix}9 & 16 \\ 8 & 17\end{bmatrix} - \begin{bmatrix}4 & 16 \\ 8 & 12\end{bmatrix} - \begin{bmatrix}5 & 0 \\ 0 & 5\end{bmatrix}$ $P^{2}-4P-5I = \begin{bmatrix}9 - 4 - 5 & 16 - 16 - 0 \\ 8 - 8 - 0 & 17 - 12 - 5\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$
Since $P^{2}-4P-5I$ results in the zero matrix, Assertion (A) is correct.
Explaining Reason (R): Cayley-Hamilton Theorem
Reason (R) states that "Every square matrix satisfies its own characteristic equation." This statement is a fundamental theorem in linear algebra known as the Cayley-Hamilton Theorem. It holds true for all square matrices.
Therefore, Reason (R) is correct.
Connecting Assertion and Reason
To understand the connection, let's find the characteristic equation for matrix $P$. The characteristic equation is given by $\det(P - \lambda I) = 0$.
For $P=\begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}$, $P - \lambda I = \begin{bmatrix}1-\lambda & 4 \\ 2 & 3-\lambda\end{bmatrix}$.
The determinant is: $\det(P - \lambda I) = (1-\lambda)(3-\lambda) - (4 \times 2)$
$= 3 - \lambda - 3\lambda + \lambda^2 - 8$
$= \lambda^2 - 4\lambda - 5$
The characteristic equation is $\lambda^2 - 4\lambda - 5 = 0$. The Cayley-Hamilton theorem (Reason R) asserts that if we substitute the matrix $P$ for $\lambda$, the equation $P^2 - 4P - 5I = 0$ must hold true. This is precisely what Assertion (A) states.
Thus, Reason (R) correctly explains Assertion (A).
Important Questions from Miscellaneous
- Which of the following scheduler/schedulers is/are also called CPU scheduler ?
(A). Short Term Scheduler
(B). Long Term Scheduler
(C). Medium Term Scheduler
(D). Asymmetric Scheduler
Choose the correct answer from the options given below: - A situation where two or more processes are blocked, waiting for resources held by each other is called:
- External fragmentation occurs ________.
- Which disk scheduling algorithm looks for the track closest to the current head position?
- Which CPU scheduling algorithm prefers the process with the shortest burst time?
