#self doubt

Question

How to solve this question?

Answer 1

Eigen value of matrix X are -2 and -3

Eigen value of Identity matrix are 1 and 1

Now Eigen value for matrix $(X + I)^{-1}(X+5I)$ are

= $(-2 + 1)^{-1}(-2+5)$ and $(-3 + 1)^{-1}(-3+5)$

= $-1*3$ and $\frac{-1}{2}*2$

= $-3$ and $-1$.

Comments

src: https://www.geeksforgeeks.org/eigen-values-and-eigen-vectors/

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Nice answer. One thing might be mentioned that if matrix $A$ has Eigen value $\lambda$ and $B$ has Eigen value $\mu$ then matrix $AB$ has Eigen value $\lambda \mu$ if matrices $A,B$ and $AB$ share the same set of Eigen vectors..It can be proved easily...Here, matrices $(x+I)^{-1}$ and $(x+5I)$ always have the same set of Eigen vectors. Matrices $x,(x+I),(x+I)^{-1},(x+5I),(x+I)^{-1}(x+5I)$ all have the same set of Eigen vectors..This might be important because for similar kind of questions, there may give another matrix $B$ with Eigen value $\lambda$ and ask about the Eigen value $(x+I)^{-1}B$.

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@ankitgupta.1729 yes my doubt is there only..I am asking him that only..same goes for addition also..that they must share common eigen vectors..then only we can add their eigen value