The Gateway to Computer Science Excellence

+21 votes

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

0

In 3rd matrix. ..if we apply R2 -> R2-R1 then it transforms into upper triangular matrix and so getting eigen values 1 and 2 ...where am I wrong here.??

0

3rd matrix where a11= 1 ,a12 = -1,a21=1,a22=1.

After R2->R2-R1

a21-a11=1-1=0

a22-a12=1-(-1)=2

SO it is upper triangular..

eigen values are diagonal entries..

So 1 and 2 are eigen values...isn't it.??

After R2->R2-R1

a21-a11=1-1=0

a22-a12=1-(-1)=2

SO it is upper triangular..

eigen values are diagonal entries..

So 1 and 2 are eigen values...isn't it.??

0

I think 1 & 2 will be the eigen values of $\begin{bmatrix} 1 & -1\\ 0& 2 \end{bmatrix}$ and not $\begin{bmatrix} 1 & -1\\ 1& 1 \end{bmatrix}$.

+10

If you add a row of A to another row, or exchange rows, the eigenvalues

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.

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.

+6

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

+24 votes

Best answer

0

@Keith, @Arjun, should not be for Matrix (4) eigenvalue = -1,-1 ? Please check. I think 0,-2 is not correct. IT does not change the final answer, someone crosscheck. λ

+1

+24 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.

52,345 questions

60,497 answers

201,862 comments

95,319 users