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GATE 2020 ST – Question 38

Matrix Theory · Eigenvalues and Matrix Decompositions · 2 marks · Multiple choice

For real a,b,c, let $$M=\begin{pmatrix}a&ac&0\\1&c&0\\b&bc&1\end{pmatrix}.$$ Which is true?

  1. Rank(M)=3 for every a,b,c.
  2. If a+c=0, M is diagonalizable for every b.
  3. M always has a pair of orthogonal eigenvectors.
  4. If b=0 and a+c=1, M is not idempotent.

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Correct answer: (C) M always has a pair of orthogonal eigenvectors.

Explanation

The nonzero vector v=(−c,1,0) satisfies Mv=0 for every parameter choice. Also w=(0,0,1) satisfies Mw=w. Their dot product is zero. Hence an orthogonal eigenvector pair always exists, C. The displayed zero-eigenvalue vector also refutes full rank.

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