GATE CS 2015 Set 1 Q.46 (CS Q.36): Linear Algebra

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}

$a = 6, b = 4$

$a = 4, b = 6$

$a = 3, b = 5$

$a = 5, b = 3$

Official answer and solution

Official answer: D
From: Official GATE 2015 answer key

Solution

Relate the trace and determinant of matrix A to the sum and product of its eigenvalues.

Given $A = \begin{pmatrix} 1 & 4 \\ b & a \end{pmatrix}$ with eigenvalues $\lambda_1 = -1$ and $\lambda_2 = 7$ .

Trace property: The trace of a matrix equals the sum of its eigenvalues:
$\text{Trace}(A) = 1 + a = \lambda_1 + \lambda_2 = -1 + 7 = 6$ $1 + a = 6 \implies a = 5.$
Determinant property: The determinant equals the product of its eigenvalues:
$\det(A) = 1 \cdot a - 4 \cdot b = \lambda_1 \cdot \lambda_2 = (-1)(7) = -7$
Substitute $a = 5$ :
$5 - 4b = -7 \implies 4b = 12 \implies b = 3.$

Thus, $a = 5$ and $b = 3$ (option D).

Final answer

Common trap

Transposing entries of A when writing the characteristic equation.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 1
Question: 46 of 65, printed as CS Q.36
Section: Computer Science and Information Technology

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