Step 1

\(\displaystyle{A}={\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{102}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&{51}\end{array}\right]}\)

In this problem, we have to find matrices L(lower triangular) and U(upper triangular) for which A=LU

Initial Matrix: \(\displaystyle{\left[\begin{array}{ccccc} {1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{1}\end{array}\right]}{\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{102}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&{51}\end{array}\right]}\)

Using some row transformations we convert this initial matrix into LU factorization,

where L will the matrix having all diagonal entries 1, and entries above diagonal 0.

Step 2

by some row transformations, we get the matrix L and matrix U

\(\displaystyle{L}={\left[\begin{array}{ccccc} {1}&{0}&{0}&{0}&{0}\\{1}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&-\frac{{1}}{{5}}&-\frac{{13}}{{10}}&{1}&{0}\\{0}&{0}&{0}&\frac{{51}}{{5}}&{1}\end{array}\right]}\)

\(\displaystyle{U}={\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{0}&-{5}&{6}&-{8}\\{0}&{0}&-{4}&{2}\\{0}&{0}&{0}&{5}\\{0}&{0}&{0}&{0}\end{array}\right]}\)

If we multiply L and U, we will get the same matrix A.

\(\displaystyle{A}={\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{102}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&{51}\end{array}\right]}\)

In this problem, we have to find matrices L(lower triangular) and U(upper triangular) for which A=LU

Initial Matrix: \(\displaystyle{\left[\begin{array}{ccccc} {1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{1}\end{array}\right]}{\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{102}&-{9}&{4}&{2}\\{0}&{0}&-{4}&{2}\\{0}&{1}&{4}&{4}\\{0}&{0}&{0}&{51}\end{array}\right]}\)

Using some row transformations we convert this initial matrix into LU factorization,

where L will the matrix having all diagonal entries 1, and entries above diagonal 0.

Step 2

by some row transformations, we get the matrix L and matrix U

\(\displaystyle{L}={\left[\begin{array}{ccccc} {1}&{0}&{0}&{0}&{0}\\{1}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&-\frac{{1}}{{5}}&-\frac{{13}}{{10}}&{1}&{0}\\{0}&{0}&{0}&\frac{{51}}{{5}}&{1}\end{array}\right]}\)

\(\displaystyle{U}={\left[\begin{array}{cccc} {102}&-{4}&-{2}&{10}\\{0}&-{5}&{6}&-{8}\\{0}&{0}&-{4}&{2}\\{0}&{0}&{0}&{5}\\{0}&{0}&{0}&{0}\end{array}\right]}\)

If we multiply L and U, we will get the same matrix A.