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Let $A$ be a square matrix and $(\text{adj} A)$ be its adjoint, show that the eigen values of matrices $A$. $(\text{adj} A)$ and $A\cdot(\text{adj} A)$ are real.

I tried to solve by using the equation $A\cdot(\text{adj} A) = |A|\cdot I = A\cdot(\text{adj} A)$, then the eigen values are coming to be $|A|$.

How to say $|A|\in\mathbb{R}$?

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  • $\begingroup$ You do have the problem that if $A$ doesn't have full rank, its determinant is zero, but the adjoint/adjugate still makes sense. $\endgroup$ – Chappers Sep 26 '15 at 13:07
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This is not true. To see this, pick $A = \begin{pmatrix} i & 1 \\ 0 & 1 \end{pmatrix}$. If you restrict the elements of $A$ to be real, however, the statement is true.

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  • $\begingroup$ yeah you are correct .But the question was asked in an exam.If elements of A are real , |A| will definitely be real and the statement is true $\endgroup$ – Balaji Sep 26 '15 at 12:58
  • $\begingroup$ Then you should consider contacting your teacher:) $\endgroup$ – Lionel Ricci Sep 26 '15 at 13:00

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