GATE CS 2019 Q.54 (CS Q.44): Linear Algebra

Consider the following matrix:

R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}

The absolute value of the product of Eigen values of $R$ is ________.

Your answer

Official answer and solution

Solution

Recognize the Vandermonde matrix structure whose determinant equals the product of eigenvalues.

A fundamental property of square matrices is that the product of the eigenvalues of a matrix $R$ is equal to its determinant:

\prod_{i=1}^n \lambda_i = \det(R)

The matrix $R$ is given by:

R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}

Each row $i$ has the form $[1, x_i, x_i^2, x_i^3]$ with $x_1 = 2, x_2 = 3, x_3 = 4, x_4 = 5$ .

This is a Vandermonde matrix. The determinant of an $n \times n$ Vandermonde matrix is given by the product of all pairwise differences:

\det(R) = \prod_{1 \le i < j \le 4} (x_j - x_i)

Evaluating each pairwise difference:

$(x_2 - x_1) = 3 - 2 = 1$
$(x_3 - x_1) = 4 - 2 = 2$
$(x_4 - x_1) = 5 - 2 = 3$
$(x_3 - x_2) = 4 - 3 = 1$
$(x_4 - x_2) = 5 - 3 = 2$
$(x_4 - x_3) = 5 - 4 = 1$

Multiplying the terms:

\det(R) = 1 \times 2 \times 3 \times 1 \times 2 \times 1 = 12

The absolute value of the product of eigenvalues is $|12| = 12$ .

Final answer

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

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.