GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 23

Question

If the eigen value of a $2 \times 2$ matrix $A$ are $-3$ and $-4,$ the sum of the eigen value of the matrices $(A + 2I)^{-1}$ and $(A + 8I)^{2}=$ ______

Comments

https://gateoverflow.in/blog/8877/linear-algebra-important-points

Answer 1

If $\lambda$ is an eigen value of a matrix $A$ then,

• $k\lambda$ is an eigen value of $kA.$
• $\lambda^{n}$ is an eigen value of $A^{n}.$
• $\dfrac{1}{\lambda}$ is an eigen value of $A^{-1}.$
• $\lambda \pm k$ is an eigen value of $A \pm kI,$ where $k$ is an any scalar value.
• $\dfrac{1}{\lambda \pm k}$ is an eigen value of $(A \pm kI)^{-1},$ where $k$ is an any scalar value.
• $(\lambda \pm k)^{2}$ is an eigen value of $(A \pm kI)^{2},$ where $k$ is an any scalar value.
• $\dfrac{\mid A \mid}{\lambda}$ is an eigen value of $\text{Adj}\;A$.
• $a_{0}\lambda^{2} + a_{1} \lambda + a_{2}$ is an eigen value of $a_{0}A^{2} + a_{1} A + a_{2} I$.
• Eigenvalues of upper triangular (or) lower triangular (or) diagonal (or) scalar (or) identity matrix are the principal diagonal elements itself.
• If $A$ is involutary matrix $(A^{2} = I)$ of even order, then $\text{trace(A) is even}$.

Now, Eigen value of $(A + 2I) = -3 + 2 = -1$

$\implies$Eigen value of $ (A + 2I)^{-1} = \dfrac{1}{-1} = -1$

And, eigen value of $(A + 2I) = -4 + 2 = -2$

$\implies$ Eigen value of $ (A + 2I)^{-1} = \dfrac{1}{-2} = -0.5$

Again, Eigen value of $(A + 8I) = -3 + 8 = 5$

$\implies$ Eigen value of $ (A + 8I)^{2} = 5^{2} = 25$

And,  Eigen value of $(A + 8I) = -4 + 8 = 4$

$\implies$ Eigen value of $ (A + 8I)^{2} = 4^{2} = 16$

$\therefore$ Sum of the eigen values of $(A + 2I)^{-1}$ and $(A + 8I)^{2} = -1 + -0.5 +25 + 16=39.5$

So, the correct answer is $39.5.$