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GATE 2023 CE (CE2) – Question 38

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

Cholesky decomposition is carried out on the following square matrix $[A]=\begin{bmatrix}8&-5\\-5&a_{22}\end{bmatrix}$. Let $l_{ij}$ and $a_{ij}$ be the (i,j)th elements of matrices [L] and [A], respectively. If the element $l_{22}$ of the decomposed lower triangular matrix [L] is 1.968, what is the value (rounded off to the nearest integer) of the element $a_{22}$?

  1. 5
  2. 7
  3. 9
  4. 11

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Correct answer: (B) 7

Explanation

**Cholesky decomposition** writes $A=LL^T$ with $L$ lower triangular. For a $2\times2$ matrix:
$$L=\begin{bmatrix}l_{11}&0\\l_{21}&l_{22}\end{bmatrix},\quad A=\begin{bmatrix}l_{11}^2&l_{11}l_{21}\\l_{11}l_{21}&l_{21}^2+l_{22}^2\end{bmatrix}.$$

**Step 1: first column.**
$$l_{11}=\sqrt{8}=2.828,\qquad l_{21}=\frac{-5}{l_{11}}=\frac{-5}{2.828}=-1.768 .$$

**Step 2: $a_{22}$ from the (2,2) entry.**
$$a_{22}=l_{21}^2+l_{22}^2=\frac{25}{8}+(1.968)^2=3.125+3.873=6.998\approx\mathbf{7}.$$

Answer **B**.