Two matrix P and Q is given below:  Matrix P = \\begin{pmatrix}1&2&3\\\\ 3&2&1\\\\ 2&3&a\\end{pmatrix} ;  Matrix Q = \\begin{pmatrix}3&4&7\\\\ 6&b&1\\\\ 0&3&1\\end{pmatrix} Matrix P + Q has one eigenvalue equal to 13. The sum of the other two eigenvalues is ___________ . (A) a+b-2 (B) a+b+6 (C) a+b (D) None of these

Answer: (D)

Explanation:  P+Q=\\begin{pmatrix}4&6&10\\\\ 9&2+b&2\\\\ 2&6&1+a\\end{pmatrix}  Let λ1 + λ2 are the other two eigen values. Sum of eigenvalues is equal to the trace of the matrix. So, 4 + 2+b + 1+a= 13 + λ1 + λ2 7 + a + b= 13 + λ1 + λ2 Hence, λ1 + λ2 = a + b – 6 Option (D) is correct.

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  • Last Updated : 20 Jan, 2019

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