Let A = LU, where L is an invertible lower triangular

Solution 12E

While finding the LU factorization of a matrix A, the matrix A is reduced to echelon form by elementary row operations, at each step the pivotal column of A is noted down and these pivotal columns form the L matrix. As the main diagonal of L consists of only 1’s so if the pivotal entry in the first column of A is a number other than 1, the whole column is divided by that number, so as to get the first entry of the first column of L as 1. That is the reason why the first column of A is a multiple of first column of L.

After the first elementary row operations by which all the entries in the first column of A below the pivotal entry are made zero the equivalent matrix is obtained, the second column of this equivalent matrix becomes the second column of L, with the first entry as zero and second entry that is the pivotal entry as 1. Thus the second column of A is row reduced to the second column of L.

For example consider the LU factorization of the below matrix .