Bis said to be column equivalent to A. If B=P AQ, then Bis said to be equivalent
to to the matrix A.
All elementary matrices have inverses and are therefore nonsingular matrices. If
Ais nonsingular, then A−^1 exists. For Anonsingular, one can perform a sequence of
elementary row transformations on the matrix Aand reduce Ato an identity matrix.
These operations are denoted by
EkEk− 1 ···E 3 E 2 E 1 A=I or P A =I (10 .24)
Right-multiplication of equation (10.24) by A−^1 gives
P=EkEk− 1 ···E 3 E 2 E 1 =A−^1. (10 .25)This equation suggests how one might build a “machine” for finding the inverse
matrix of a nonsingular matrix A. Write down the matrix An×n and append to the
right of it the identity matrix In×n.By doing this the identity matrix can then be
used like a “recording device,” to record all elementary row operations that are
performed on A. That is, whatever a row operation is performed upon Ayou must
also perform the same row operation on the appended identity matrix. The matrix
Awith the identity matrix appended to its right-hand side is called an augmented
matrix. After writing down
A|I, (10 .26)observe that if an elementary row transformation is applied to the matrix A, then
it is possible to “record” this transformation on the right-hand side of the equation
(10.26). For example, if E 1 is an elementary row transformation applied to the
augmented matrix, one obtains
E 1 A|E 1 I.By performing a sequence of elementary row transformations upon the aug-
mented matrix, given by equation (10.26), one can change the augmented matrix to
the form
Ek···E 2 E 1 A|Ek···E 2 E 1 I, (10 .27)