Question

Cholesky decomposition is carried out on the following square matrix [A]. \[ [A] = \begin{bmatrix} 8 & -5 \\ -5 & a_{22} \end{bmatrix} \] Let \( l_{ij} \) and \( a_{ij} \) be the (i,j)\textsuperscript{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 \( a_{22} \)?}

• A. 5
• B. 7
• C. 9
• D. 11

Hint:

In Cholesky decomposition, you can find the elements of the original matrix using the elements of the lower triangular matrix. Start by calculating \( l_{11} \), then use it to find \( l_{21} \) and \( l_{22} \).

Answer 1

The Correct Option is B

\\ The Cholesky decomposition of the matrix \( [A] \) is given by: \[ [A] = [L] [L]^T \] Here, the elements of \( [L] \) are \( l_{11}, l_{12}, l_{21}, l_{22} \), and the matrix \( [L] \) is: \[ [L] = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{bmatrix} \] We already know \( l_{22} = 1.968 \). From the Cholesky decomposition, we can express the elements of matrix \( [A] \) in terms of elements of matrix \( [L] \): \[ a_{11} = l_{11}^2 = 8 \Rightarrow l_{11} = \sqrt{8} = 2.828 \] Next, using the second row, first column entry of matrix \( [A] \), we get: \[ a_{21} = l_{11} l_{21} = -5 \Rightarrow l_{21} = \frac{-5}{2.828} = -1.768 \] Now, using the second row, second column entry of matrix \( [A] \), we get: \[ a_{22} = l_{21}^2 + l_{22}^2 = (-1.768)^2 + (1.968)^2 = 3.13 + 3.87 = 7 \] Thus, the value of \( a_{22} \) is 7, and the correct answer is (B).