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 $\left[\begin{array}{cc}-1 & 0 \\1 & -1 \end{array} \right]$

- one
- two
- three
- four

### 6 Comments

usually change. Elimination does not preserve the Eigen values.

A better way is to check if | A- λI| = 0, If 1 (given in the question) is the eigen value of this matrix. and the charactersitic equation satisfies. By doing so for all the matrix we find the equation satisfies for first matrix.

Hence the answer is one , ie, option A

I found this way much faster then finding the values and then checking if the eigen values.

Elementary row operations change eigenvalues of given matrix (https://socratic.org/questions/do-elementary-row-operations-change-eigenvalues)

## 3 Answers

### 8 Comments

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.