AdjA=CT=
[
4 − 2
3 1
]
.
This gives |A|= 10 so that Ais nonsingular and the inverse is given by
A−^1 =
1
10
[
4 − 2
3 1
]
As a check, verify that AA −^1 =I
Elementary Row Operations
A very useful matrix operation is an elementary row operation performed on a
matrix. These elementary row operations can be used to obtain a wide variety of
results.
An elementary row matrix E is any matrix formed from the identity matrix
I = (δij) by performing any of the following elementary row operations upon the
identity matrix.
(a) Interchange any two rows of I
(b) Multiplication of a row of Iby any nonzero scalar m
(c) Replacement of the ith row of Iby the sum of the ith row and mtimes the jth
row, where i=jand mis any scalar.
An elementary column matrix Eis obtained if column operations are used instead
of row operations. An elementary transformation of a matrix Ais the multiplication
of Aby an elementary row matrix.
Example 10-22. Consider the matrix A =
a b c
d e f
g h i
and the elementary
matrices
E 1 where row 1 and 2 of the identity matrix are interchanged.
E 2 where row 1 is interchanged with row 3 and then rows 1 and 2 are interchanged.
E 3 where row 1 of the identity matrix is multiplied by 3.
E 4 where row 2 of the identity matrix is multiplied by 3 and the result added to row 1.
These elementary matrices can be represented
E 1 =
0 1 0
1 0 0
0 0 1
, E 2 =
0 1 0
0 0 1
1 0 0
, E 3 =
3 0 0
0 1 0
0 0 1
, E 4 =
1 3 0
0 1 0
0 0 1
,
