Question

Let C and D are two Non-singular matrices of same order and I is Identity matrix of the same order as of C and D. Then det(I-CD) is



I know answer but I want explaination give explaination.​

Answer 1

Let C and D are two Non-singular matrices of same order and I is Identity matrix of the same order as of C and D. Then det(I-CD) is

This is my approach to the question. I am not sure whether it is a correct one.

Let x be the eigenvalue of the matrix CD with the eigenvector v. Then CDv=xv. Now let y=Dv. Then DCy=DCDv=xDv=xy. This shows that y is the eigenvector of DC with the eigenvalue x.

Now DCy=xDv is non-zero if and only if x and Dv are non-zero. This implies that DC and CD have the same non-zero eigenvalues, which also means that ϕ(CD,x)=ϕ(DC,x) provided x≠0. Considering the case where x=1 we have the required results, because det(I−CD)=ϕ(CD,1)=ϕ(DC,1)=det(I−DC)

Answer 2

Answer:

mi intro please

Step-by-step explanation:

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