Question

The matrix \([A]=\begin{bmatrix}2&1\\\ 4&-1\end{bmatrix}\) is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
1. \(\begin{bmatrix}1&0\\\ 4&-1\end{bmatrix}\) and \(\begin{bmatrix}1&1\\\ 0&-2\end{bmatrix}\)
2. \(\begin{bmatrix}2&0\\\ 4&-1\end{bmatrix}\) and \(\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}\)
3. \(\begin{bmatrix}1&0\\\ 4&1\end{bmatrix}\) and \(\begin{bmatrix}2&1\\\ 0&-1\end{bmatrix}\)
4. \(\begin{bmatrix}2&0\\\ 4&-3\end{bmatrix}\) and \(\begin{bmatrix}1&0.5\\\ 0&1\end{bmatrix}\)
5.

Answer 1

Correct Answer - Option 4 : \(\begin{bmatrix}2&0\\\ 4&-3\end{bmatrix}\) and \(\begin{bmatrix}1&0.5\\\ 0&1\end{bmatrix}\)

We know that matrix A is equal to the product of lower triangular matrix L and upper triangular matrix U,

A = [L][U]

Let's check option wise,

Option 1:

L = ​\(\begin{bmatrix}1&0\\\ 4&-1\end{bmatrix}\)and U = \(\begin{bmatrix}1&1\\\ 0&-2\end{bmatrix}\)

A = [L][U] = \(\begin{bmatrix}1&1\\\ 4&6\end{bmatrix}\)

Option 2:

L = \(\begin{bmatrix}2&0\\\ 4&-1\end{bmatrix}\) U = \(\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}\)

A = [L][U] = ​ \(\begin{bmatrix}2&2\\\ 4&3\end{bmatrix}\)

Option 3:

L = \(\begin{bmatrix}1&0\\\ 4&1\end{bmatrix}\) U = \(\begin{bmatrix}2&1\\\ 0&-1\end{bmatrix}\)

A = [L][U] = ​\(\begin{bmatrix}2&1\\\ 8&3\end{bmatrix}\)

Option: 4

L =\(\begin{bmatrix}2&0\\\ 4&-3\end{bmatrix}\) U = \(\begin{bmatrix}1&.5\\\ 0&1\end{bmatrix}\)

A = [L][U] = ​\(\begin{bmatrix}2&1\\\ 4&-1\end{bmatrix}\)