Question

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).

Hint:

Use \(l_{11}=\sqrt{a_{11}}\), \(l_{21}=a_{21}/l_{11}\), \(l_{22}=\sqrt{a_{22}-l_{21}^2}\).

Answer 1

Correct Answer: 5

Step 1: Understanding the Question.
Cholesky decomposition expresses a symmetric positive-definite matrix \(A\) as \(A = LL^T\), where \(L\) is a lower triangular matrix. We need the magnitude of the entry \(l_{22}\) of this \(L\) for the given matrix.

Step 2: Key Formula or Approach.
For a \(2\times2\) matrix \(A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}\), writing \(L = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22}\end{bmatrix}\) and expanding \(LL^T\) gives:
\[ l_{11} = \sqrt{a_{11}}, \quad l_{21} = \frac{a_{21}}{l_{11}}, \quad l_{22} = \sqrt{a_{22} - l_{21}^2} \]

Step 3: Detailed Explanation.
Here \(a_{11} = 9\), \(a_{21} = 15\), \(a_{22} = 50\).
First find \(l_{11}\):
\[ l_{11} = \sqrt{9} = 3 \]
Next find \(l_{21}\):
\[ l_{21} = \frac{15}{3} = 5 \]
Now find \(l_{22}\):
\[ l_{22} = \sqrt{50 - 5^2} = \sqrt{50-25} = \sqrt{25} = 5 \]
We can double check by reconstructing \(LL^T\):
\[ L = \begin{bmatrix} 3 & 0 \\ 5 & 5\end{bmatrix}, \quad LL^T = \begin{bmatrix} 9 & 15 \\ 15 & 50\end{bmatrix} \]
which matches the given matrix \(A\), confirming the decomposition is correct.

Step 4: Final Answer.
\(|l_{22}| = 5\).
\[ \boxed{5} \]