In the LU decomposition of the matrix \(\begin{bmatrix}2 & 2 \\ 4 &…
2015
In the LU decomposition of the matrix \(\begin{bmatrix}2 & 2 \\ 4 & 9\end{bmatrix}\), if the diagonal elements of \(U\) are both 1, then the lower diagonal entry \(l_{22}\) of \(L\) is________.
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Correct answer: 5
Key idea: write L and U with the given constraint that U has ones on its diagonal, multiply them, and equate entries to the original matrix.
Let L = [[a, 0], [b, c]] and U = [[1, d], [0, 1]].
Multiply: L · U = [[a, a·d], [b, b·d + c]].
Equate to the given matrix [[2, 2], [4, 9]] to get the system: a = 2, a·d = 2, b = 4, b·d + c = 9.
Solve: from a = 2 and a·d = 2 ⇒ d = 1. From b = 4 and b·d + c = 9 ⇒ 4·1 + c = 9 ⇒ c = 5.
Answer: l_{22} = 5