GATE 2016 EC (EC2) – Question 11
The value of $x$ for which the matrix
$$A=\begin{bmatrix}3&2&4\\9&7&13\\-6&-4&-9+x\end{bmatrix}$$
has zero as an eigenvalue is ________.
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Show answer and explanation
Correct answer: 0.95 to 1.05
Explanation
Zero is an eigenvalue exactly when $\det A=0$. Replace row 2 by row 2 minus three times row 1, giving $(0,1,1)$. Replace row 3 by row 3 plus twice row 1, giving $(0,0,x-1)$.
These operations preserve the determinant. The resulting upper triangular matrix has determinant $3(x-1)$. Thus $3(x-1)=0$, giving **$x=1$**.