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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?

1. $1, 4, 3$
2. $3, 7, 3$
3. $7, 3, 2$
4. $1, 2, 3$
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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 )

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)
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Eigen values of Identity/ Diagonal/ Scalar/ Upper Triangular/ Lower Triangular matrix are equal to the principle diagonal elements.
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 $\lambda =1,3,4$
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You have given proof of above property.. :)
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Sum of eigenvalues is the trace(diagonal sum) of a matrix only option a) satisfy this property

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

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