Question

The characteristic equation associated with the matrix $\left[\begin{matrix}0& 0& 3\\ 1& 0& 2\\ 0& 1& 1\end{matrix}\right]$ is

• A. $\lambda^{3} - \lambda^{2} - 2\lambda - 3 = 0$
• B. $\lambda^{3} - \lambda^{2} + 2\lambda - 3 = 0$
• C. $\lambda^{3} - \lambda^{2} - 3\lambda - 3 = 0$
• D. $\lambda^{3} - \lambda^{2} + 3\lambda - 3 = 0$

Hint:

For a $3 \times 3$ matrix, the equation is $\lambda^3 - Tr(A)\lambda^2 + (S_{11}+S_{22}+S_{33})\lambda - \det(A) = 0$, where $S_{ii}$ are principal minors.

Answer 1

The Correct Option is A

Step 1: Concept
The characteristic equation of a matrix is obtained by solving \[ \det(A-\lambda I)=0. \]

Step 2: Meaning
For the given matrix, \[ \det\begin{pmatrix} -\lambda & amp; 0 & amp; 3\\ 1 & amp; -\lambda & amp; 2\\ 0 & amp; 1 & amp; 1-\lambda \end{pmatrix}=0. \] 

Step 3: Analysis
Expanding along the first row, \[ -\lambda\left[(-\lambda)(1-\lambda)-2\right]-0 +3\left[(1)(1)-0\right]=0. \] Simplifying, \[ -\lambda(-\lambda+\lambda^2-2)+3=0, \] \[ \lambda^2-\lambda^3+2\lambda+3=0. \] Multiplying both sides by \(-1\), \[ \lambda^3-\lambda^2-2\lambda-3=0. \] 

Step 4: Conclusion
This matches the characteristic polynomial given in option (A). 

Final Answer: (A)