Elementary matrices: Let e denote an elementary row operation and let e(A) denote the result of applying the operation e to matrix. Now let E be the matrix obtained by applying to the identity matrix I, that is,
E = e(I)
Then E is called the elementary matrix corresponding to the elementary row operation e.
Note that E is always a square matrix.
Application to finding the inverse of an n×n matrix
The following algorithm finds the inverse of a matrix
Algorithm: The input is a square matrix A. The output is the inverse of A or that the inverse does not exist.
Step 1: From the n×2n (block) matrix M= [A,I], where A is the left half of M and the identity matrix I is the right half of M.
Step 2: Row reduce M to echelon form. If the process generate a zero row in the A half of M, then
A has no inverse.( Otherwise A is an triangular form).
Step 3: Further row reduce to M to its canonical form M~[I,B]
Where the identity matrix I has replaced A in the left half of M.
Step 4: Set A-1=B, the matrix that is now in the right half of M.
The justification for the above algorithm is as follows. Suppose A is invertible and, say, the sequence of elementary row operations e1, e2, —-, eq applied to M = [A,I] reduce the left half of M, which is A, to the identity matrix I. Let Ei be the elementary matrix corresponding to the operation ei. Then, by applying
[“ Let e be an elementary row operation and let E be the corresponding m×n elementary matrix. Then e(A) = EA; where A is any m×n matrix”.]
and we get
Eq, ———-, E2, E1A = I or, (Eq, ———-, E2, E1I) A=I,
so A‑1= Eq, ———-, E2, E1I
that is, A-1 can be obtained by applying the elementary row operations
e1, e2, —-, eq to the identity matrix I, which appears in the right half of M. Thus B= A-1, as claimed.
Elementary column operations: Now let A be a matrix with columns C1, C2, ….., Cn. The following operation on A, analogous to the elementary row operations, are called elementary column operations:[F1] (Column interchange): Interchange column Ci and Cj. [F1] (Column scaling): Replace Ci by kCi (where k≠0) [F1] (Column addition): Replace Cj by kCi+Cj
We may indicate each of the column operations by writing, respectively,
(1) Ci⟷Cj (2) kCi⟶Cj (3) (kCi+Cj ) ⟶Cj
Moreover each column operation has an inverse operation of the same type, just like the corresponding row operation.
Now let f denote an elementary column operation, and let F be the matrix obtained by applying f to the identity matrix I, that is,
F = f(I)
Then F is called the elementary matrix corresponding to the elementary column operation f. Note that F is always a square matrix.