GATE 2022 ST – Question 36

Matrix Theory · Inner Products and Quadratic Forms · 2 marks · Multiple choice

A symmetric 3×3 matrix M has eigenvalues −1,1,2 and corresponding unit eigenvectors u,v,w. If Mx=u+2(v+w) and M²y=u−(v+2w),find ||x+y||².

1.25
0.25
0.75
1

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Correct answer: (A) 1.25

Explanation

Distinct eigenvalues of a symmetric matrix give an orthonormal basis. Invert the eigenvalues to obtain x=−u+2v+w and y=u−v−w/2. Thus x+y=v+w/2,and squared norm=1+1/4=**1.25, A**.

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