#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

@adad20 wrong option c is given answer

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@samarpita Yeah I forgot to take inverse of $(-3 +1)$ that would be $\frac{-1}{2}$ and then the answer would be -3 and -1. I edited the answer

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why you are multiplying the eigen values...is there any rule?

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Yeah these are the properties of eigen value.

Refer: https://math.mit.edu/~gs/linearalgebra/ila0601.pdf

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(−2+1)^−1(−2+5) and (−3+1)^−1(−3+5)

= −1∗3 and −1/2∗2

How you are doing this @adad20?

 

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I am using the proerties of Eigen values for example if eigen value of matrix A is x and eigen value of matrix B is y then eigen value of matrix (A+B) is x+y similary for $A^{-1}$ it will be $x^{-1}$, these are properties of eigen values its basic concepts if you have not gone through this then please refer above pdf or you can also watch some videos on this topic. You can also try deriving these properties.

PS: Above properties hold only when A and B share the same set of eigen vectors.

Thanks @ankitgupta.1729 sir for pointing out

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Where this above mentioned property is there? I haven't got anywhere.. Can you please attach the screen shot of that page?

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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