Find LU-decomposition of the following matrix.𝐴 = [2 01 2].

The LU decomposition of a matrix is a factorization that represents the original matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U).

Given the matrix A = [2 0; 1 2], we want to find matrices L and U such that A = LU.

Step 1: Initialize L as an identity matrix and U as the matrix A.

L = [1 0; 0 1] U = [2 0; 1 2]

Step 2: For each column of U, starting from the left, subtract a multiple of that column from all columns to the right, such that all elements below the diagonal in the leftmost column are zero.

In this case, the first column of U is already in the correct form, so we move to the second column. We want to subtract a multiple of the first column from the second column such that the element below the diagonal in the second column is zero. However, this element is already zero, so no action is needed.

Step 3: For each column where we subtracted a multiple in step 2, add that multiple to the corresponding position in the L matrix.

In this case, we didn't need to subtract any multiples, so L remains as the identity matrix.

Therefore, the LU decomposition of A is:

L = [1 0; 0 1] U = [2 0; 1 2]

So, A = LU = [1 0; 0 1] * [2 0; 1 2] = [2 0; 1 2], which is the original matrix A.