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Elementary matrices| Linear Algebra

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

STOP

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) CiCj   (2) kCiCj (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.

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