Question

We have discussed elementary elimination matrices M_k in class. Prove the following two properties of elementary elimination matrices, which are very important for making LU factorization work efficiently in practice: M_k is nonsingular. Represent M_k^-1 explicitly and show that M_kM_k^-1 = M_k^-1M_k = I. The product of two elementary elimination matrices M_k and M_j with k notequalto j is essentially their "union"; and therefore they can be multiplied without any computational cost.

Answer 1

Please find the below pictures describing the answers written in details.

According to the lemma, Every elementary elimination matrix is invertible and the inverse is again an elementary matrix. If, an elementary matrix M_k is obtains from I by using a certain row - operation q, Then, M^-1_k is obtained from I by the inverse operation q^-1 defined as follows: (i) If, q is the adding operation, then q^-1 is also an adding operation (ii) If, q is the multiplication of a row by x, then, q^-1 is the multiplication of the same row by x^-x (iii) If, q is swapping operation, Then, q^-1 = q so, for every elementary operation q if we apply q and q^-1 to I, Then, we return to I \r\n But, by the theorem about elementary matrices, application of q row operation to q matrix is equivalent to multiplying this matrix by the corresponding elementary matrix Thus, Let M_k be the elementary matrix corresponding to the operation q_1 and M^-1_k be the operation q^-1 M_k M_k^-1 = M^-1_k M_k = I

Below is the answer for part two.