Topic: Engineering Mathematics
The Algebraic multiplicity of the matrix $A=\begin{bmatrix}0&1&0\\ 0&0&1\\ 1&-3&3\end{bmatrix}is-$
- A. 1
- B. 2
- C. 3
- D. 4
Answer: C. 3
Explanation: Algebraic Multiplicity (AM) is the number of times an eigenvalue is repeated as a root of the characteristic equation $|A-\lambda I|=0$. Characteristic equation: $\begin{vmatrix}-\lambda&1&0\\ 0&-\lambda&1\\ 1&-3&3-\lambda\end{vmatrix}=0 \Rightarrow -\lambda[-\lambda(3-\lambda)-(-3)(1)] - 1(0 - 1) + 0 = -\lambda(\lambda^2 - 3\lambda + 3) + 1 = -\lambda^3 + 3\lambda^2 - 3\lambda + 1 = 0 \Rightarrow (\lambda-1)^{3} = 0$. Here, $\lambda=1$ is an eigenvalue repeated 3 times, hence algebraic multiplicity is 3.