Complex Eigenvalues

Algebra Level 5

Consider the matrix A = ( i 0 − 1 1 − i 1 0 1 2 i + 1 ) . A =\begin{pmatrix} i & 0 & -1 \\ 1 & -i & 1 \\ 0 & 1 & 2i+1 \end{pmatrix}.

If the sum of the eigenvalues is a + b i a+bi and the product of the eigenvalues is c + d i c+di , calculate a + b + c + d a+b+c+d .


The answer is 4.

Steven Zheng by Steven Zheng , 26, China

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1 solution

Thushar Mn
Apr 20, 2015

sum of eigen values =sum of dioganal elements. product of eigen values = determinent of metrix. ^_^

3 Helpful 0 Interesting 0 Brilliant 0 Confused

For those who would like to look at some work, not just a statement of what to do, but how to do it:

a + b i = (i) + (-i) + (2 i+1) = 0 + 1 + 2*i -> a = 1, b = 2

Det(A) = i{-i,1;1,2i+1} -1{0,-1;1,2i+1} + 0

Det(A) = i(2 - i - 1) - 1(-(-1))

Det(A) = i + 1 - 1

Det(A) = i -> c = 0, d = 1

a + b + c + d = 1 + 2 + 0 + 1 = 4 Which is the correct answer.

Luc-Andre Sabourin - 5 years, 9 months ago

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