Elementary Line Operation Methods and Determinants for Finding the Inverse Matrix
In this article EPandu
will explain several methods to find inverse matrix values that are larger
than 2 x 2 using elementary and determinant row operations.
Calculating Inverses with Elementary Line Operations (ELO)
A matrix n x n is called
an elementary matrix if the matrix can be obtained from a matrix of units of n
x n namely I_{n} by performing a single elementary row operation.
Example 1
Suppose: ELO (i) is
For I_{2}
For I_{3}
_{Add (3 x) Third Row to the first line}
_{}
For the I_{4} exchange the second row with the third row.
If matrix A is multiplied
to the left by an elementary matrix E, the effect is to demonstrate ELO to A.
Theorem 1.
The elementary E matrix is
produced by doing a certain ELO on I_{m} and
suppose A is a matrix m x n. Then the EA product is the same as the matrix
produced by doing the same ELO on A.
Example 2.
Calculate the A Matrix
below using Elementary Line Operations and E and A Matrix then compare the
results.
Note: E matrix is the
result of adding (3 x) the first line from I_{3} to the third row.
 Elemental Operation Line
to matrix A the result is
 Multiplication of matrix E and matrix A are
When compared between Elementary Line Operations A matrix with the results of matrix multiplication E and A, the results are the same, namely
If the ELO is applied to the unit matrix I to produce an elementary matrix E, then there is a
second row matrix operation which when applied to the E Matrix will return the I Matrix. For example, if the E matrix is obtained by
multiplying the iline from I with the
k constant not equal to zero, then I can be found again if row i of E is multiplied by 1 / k.
Operation
Line on E Matrix that produces Matrix I

Operation
Line on I Matrix that produces Matrix E

Multiply the rowi by 1 / k, where
k ≠ 0

Multiply the rowi by k, where
k ≠ 0

Exchange the iline with the jline

Exchange the iline with the jline

Add the (k) line i to the jline

Add the (k) line i to the jline

The righthand operations
of this table are called Inverse Operations from the corresponding operations
in the left section.
Example 3.
Using the results in the
Elementary Row Operations table above, the first three elementary matrix given in Example 1 can be returned to unit matrix by applying the following
operations:
 Multiply the second line in (i) with 1/3;
 Add (3 x) the third row from (ii) to the first row
 Exchange the second and fourth lines in (iii)
Theorem 2.
The A matrix has an
inverse if and only if Matrix A is the equivalent of a line to I.
Example 4.
Determine the inverse of
the matrix below,
Solution:
We will reduce matrix A to
the unit matrix using row operations and apply these operations simultaneously
to matrix I to
produce A^{1}. This can be solved by holding the unit matrix to the
right of matrix A and by applying row operations to both segments until the
left segment is reduced to matrix I. Then
the final matrix will have the form [I  A^{1}].
This calculation can be done as follows.
Note: in determining A^{1} if a zero row is
obtained on the lefthand side, then A^{1} cannot be found or A has no
inverse.
Determining Inversion by Using Determinants
Next, we will show how to
determine inverses by using determinants which in solution we need Adjoin from
the matrix. Because in the previous "Rules for CalculatingDeterminants" article we discussed how to determine the adjoining of a
matrix, then in this article the discussion will begin with the following
theorem.
If A is a matrix that can
be reversed, then:
Example 1.
Determine the inverse of matrix A below,
Solution:
By using the rules for calculating the determinants described in the Article Determining Rules, the determination of the matrix A is obtained:
det (A) = 64
and adjoin A are:
Then the A^{1}
Inverse Matrix produced is
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