The GATE Grind

GATE 2016 ME (ME2) – Question 11

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 1 mark · Multiple choice

The eigenvalues of $A=\begin{bmatrix}2&1\\1&k\end{bmatrix}$ are positive when ______.

  1. $k>1/2$
  2. $k>-2$
  3. $k>0$
  4. $k<-1/2$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $k>1/2$

Explanation

For A=[[2,1],[1,k]], positive definiteness requires positive leading principal minors. The first is2>0; determinant is2k−1, which must be positive. Hence **k>1/2, A**. Symmetry ensures real eigenvalues, so this criterion gives both positive.