GATE | GATE CS 2010 | Question 29

Consider the following matrix
A = CSE_201029
If the eigenvalues of A are 4 and 8, then

(A) x=4, y=10
(B) x=5, y=8
(C) x=-3, y=9
(D) x= -4, y=10


Answer: (D)

Explanation:  A = \begin{bmatrix}     2    &   3  \\     x     &  4  \\ \end{bmatrix} \newline \newline [A - \lambda i]=0 \newline \newline \begin{bmatrix}     2-\lambda     &   3  \\     x     &  y-\lambda  \\ \end{bmatrix} = 0 \newline \newline (2-\lambda)(y-\lambda) - 3x =0  \newline given \hspace{0.2cm} eigen\hspace{0.2cm} values: \lambda=4,8 \newline if \hspace{0.2cm} we \hspace{0.2cm} use \hspace{0.2cm} \lambda=4\hspace{0.2cm} then \hspace{0.2cm} equation, \hspace{0.2cm} -2(y-4)-3x=0 . . . . ...(i) \newline if \hspace{0.2cm} we \hspace{0.2cm} use \hspace{0.2cm} \lambda=8\hspace{0.2cm} then \hspace{0.2cm} equation, \hspace{0.2cm} -6(y-8)-3x=0 . . . . ...(ii) \newline Solving \hspace{0.2cm} equations \hspace{0.2cm} (i) & (ii), we \hspace{0.2cm} get, x=-4,y=10

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