GATE 2020 EE – Question 52
The number of purely real elements in a lower triangular representation of the given $3\times3$ matrix, obtained through the given decomposition is ________.
$$\begin{bmatrix}2&3&3\\3&2&1\\3&1&7\end{bmatrix}=\begin{bmatrix}a_{11}&0&0\\a_{12}&a_{22}&0\\a_{13}&a_{23}&a_{33}\end{bmatrix}\begin{bmatrix}a_{11}&0&0\\a_{12}&a_{22}&0\\a_{13}&a_{23}&a_{33}\end{bmatrix}^T$$
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Correct answer: (B) 6
Explanation
Solving the Cholesky-type factorization row by row: $a_{11}=\sqrt2$, $a_{12}=a_{13}=\frac{3}{\sqrt2}$ are real. Then $a_{22}^2=2-4.5<0$, so $a_{22}$ is purely imaginary, and $a_{23}=\frac{1-4.5}{a_{22}}$ is also imaginary. Finally $a_{33}^2=7-4.5+4.9$ is positive, so $a_{33}$ is real. The matrix is not positive definite, which is why the question was awarded to all candidates.