GATE Computer Science (CS) 2019 Solved Papers

Question : 54 of 65

Marks: +1, -0

Consider the following matrix:

R=[\table 1 , 2 , 4 , 8 ; 1 , 3 , 9 , 27 ; 1 , 4 , 16 , 64 ; 1 , 5 , 25 , 125 ]

The absolute value of the product of Eigen values of R is _________.

Solution:

Product of eigenvalues is same as the determinant of a matrix.

[\table 1 , 2 , 2^{2} , 2^{3} ; 1 , 3 , 3^{2} , 3^{3} ; 1 , 4 , 4^{2} , 4^{3} ; 1 , 5 , 5^{2} , 5^{2}]=[\table 1 , 2 , 2^{2} , 2^{3} ; 1 , 3-2 , 3^{2}-2^{2} , 3^{3}-2^{3} ; 1 , 4-2 , 4^{2}-2^{2} , 4^{3} -2^{3} ; 1 , 5-2 , 5^{2}-2^{2} , 5^{3}-2^{3}]

=(3-2)(4-2)(5-2)[\table 1 , 5 , 19 ; 1 , 6 , 28 ; 1 , 7 , 39]

=1.2 .3[\table 1 , 5 , 19 ; 0 , 1 , 9 ; 0 , 2 , 20]

=1.2 .3 .2=12

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