GATE CSE 2011 | Question: 40

Question

Consider the matrix as given below.

$$\begin{bmatrix} 1 & 2 & 3 \\  0 & 4 & 7 \\  0 & 0 & 3\end{bmatrix}$$

Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

• A. $1, 4, 3$
• B. $3, 7, 3$
• C. $7, 3, 2$
• D. $1, 2, 3$

Comments

Eigen values of Identity/ Diagonal/ Scalar/ Upper Triangular/ Lower Triangular matrix are equal to the principle diagonal elements. (eigenvalues are calculated for square matrix only)

Their determinant is the product of their principal diagonal elements. (for square matrix )

Answer 1

The answer is $A.$

The given matrix is an upper triangular matrix and the eigenvalues of upper or lower triangular matrix are

the diagonal values itself.(Property)

Comments

Eigen values of Identity/ Diagonal/ Scalar/ Upper Triangular/ Lower Triangular matrix are equal to the principle diagonal elements.

Answer 2

another method is also very shortcut $\begin{vmatrix} 1-\lambda &2 &3 \\ 0 &4-\lambda &7 \\ 0& 0 &3-\lambda \end{vmatrix}$

now to find Eigen values we calculate determinant of matrix and we calculate determinant along 1st column which give

$(1-\lambda )(4-\lambda )(3-\lambda )=0$    which gives $\lambda =1,3,4$

Comments

You have given proof of above property.. :)

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@saket nandan Please correct your answer. Eigenvalues should be 1,4,3 not 1,2,4.

Answer 3

it should follow two conditions 

1-Trace of matrix = sum of Eigen Values =sum of diagonal

Trace=1+4+3=8

2 -Determinant of given matrix = 12 

As we know product of eigen values is determinant of matrix 

Now we have to check options 

(A) 1, 4, 3 =1+4+3=8 satisfied

(B) 3, 7, 3 =3+7+3=13 Wrong

(C) 7, 3, 2 =7+3+7=17 Wrong

(D) 1, 2, 3 =1+2+3=6 Wrong

So option A is answer

Answer 4

Sum of eigenvalues is the trace(diagonal sum) of a matrix only option a) satisfy this property

Answer 5

Eigen values of Identity/ Diagonal/ Scalar/ Upper Triangular/ Lower Triangular matrix are equal to the principle diagonal elements.

 

or 

 

Comments

option A is correct because

 of the lower trangular are zeros so the diagonal of the matrix will be the eigen values

 

Answer 6

Answer: A 

• Product of eigenvalues equals the Determinant of the matrix.

$$\begin{vmatrix}
1 & 2 & 3 \\

0 & 4 & 7\\
0 & 0 & 3
\end{vmatrix} =12$$

$$1.4.3=12$$