GATE CS 2014 Set 2 Q.57 (CS Q.47): Linear Algebra

The product of the non-zero eigenvalues of the matrix

\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}

is _____.

Your answer

Official answer and solution

Solution

Decompose the block-symmetric matrix to find its eigenvalues and calculate the product of all non-zero eigenvalues.

Matrix Structure:
$M = \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}.$
Block Decomposition:
By permuting rows and columns to group indices $(1, 5)$ and $(2, 3, 4)$ , $M$ is block-diagonal:
$M \sim \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix}$
where
$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}_{2 \times 2}, \qquad B = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}_{3 \times 3}.$
Eigenvalues of Block $A$ :
$A$ is an all-ones $2 \times 2$ matrix.
- Rank = 1, so one eigenvalue is $0$ .
- Trace = $1 + 1 = 2$ , so the other eigenvalue is $2$ .
  Eigenvalues of $A$ : $\{2, 0\}$ .
Eigenvalues of Block $B$ :
$B$ is an all-ones $3 \times 3$ matrix.
- Rank = 1, so two eigenvalues are $0$ .
- Trace = $1 + 1 + 1 = 3$ , so the third eigenvalue is $3$ .
  Eigenvalues of $B$ : $\{3, 0, 0\}$ .
Product of Non-zero Eigenvalues:
The complete set of eigenvalues of $M$ is $\{2, 3, 0, 0, 0\}$ .
The non-zero eigenvalues are $2$ and $3$ .
$\text{Product} = 2 \times 3 = 6.$

Final answer

Common trap

Trying to compute the 5x5 characteristic polynomial directly without exploiting block independence or rank-1 properties.

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