GATE 2026 CE (CE1) – Question 27
A matrix is given as
$$\begin{bmatrix} 9 & 15 \\ 15 & 50 \end{bmatrix}$$
By performing Cholesky decomposition, $|l_{22}|$ of the lower triangular matrix is ______ (*in integer*).
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Correct answer: 5
Explanation
Write the matrix as $LL^T$ with $L = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{bmatrix}$. Then $l_{11}^2 = 9$ gives $l_{11} = 3$, and $l_{11}l_{21} = 15$ gives $l_{21} = 5$. Finally $l_{21}^2 + l_{22}^2 = 50$ gives $l_{22}^2 = 25$, so $|l_{22}| = 5$.