The GATE Grind

GATE 2015 CS – Question 59

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Consider the following $2 \times 2$ matrix $A$ where two elements are unknown and are marked by $a$ and $b$. The eigenvalues of this matrix are $-1$ and $7$. What are the values of $a$ and $b$?

$$A = \begin{pmatrix} 1 & 4 \\ b & a \end{pmatrix}$$

  1. $a = 6, b = 4$
  2. $a = 4, b = 6$
  3. $a = 3, b = 5$
  4. $a = 5, b = 3$

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Show answer and explanation

Correct answer: (D) $a = 5, b = 3$

Explanation

The sum of the eigenvalues equals the trace, so $1 + a = -1 + 7 = 6$ and $a = 5$. The product of the eigenvalues equals the determinant, so $a - 4b = -7$, which gives $5 - 4b = -7$ and $b = 3$.