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31 votes
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How many of the following matrices have an eigenvalue 1?
$\left[\begin{array}{cc}1 & 0 \\0 & 0 \end{array} \right]\left[\begin{array}{cc}0 & 1 \\0 & 0 \end{array} \right] \left[\begin{array}{cc}1 & -1 \\1 & 1 \end{array} \right]$ and $\left[\begin{array}{cc}-1 & 0 \\1 & -1 \end{array} \right]$

  1. one
  2. two
  3. three
  4. four

3 Answers

Best answer
32 votes
32 votes
Characteristic equation is $ | A- \lambda I | = 0 $
(1)$$\begin{vmatrix}1- \lambda & 0\\
0 & - \lambda\\
\end{vmatrix}  = 0$$$ (1- \lambda )(- \lambda) = 0 $
$\implies \lambda =0,1 $

Similarly, (2)  $ \lambda =0,0 $

(3) $ \lambda \neq 1 $

(4) $ \lambda =-1,-1 $

Therefore, Answer is (A) one
selected by
31 votes
31 votes

Out of given answers, 3 of them are (upper/lower) triangular matrices. So, that will give eigen values directly.

Property: The eigen value of triangular / unity / scalar / diagonal matrices is equal to principal diagonal elements itself.

3rd option need to be solved which wont be equal to 1.

So first matrix only have eigen value 1.

Answer:

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