Transcript Properties of Inverse Matrices

Properties of Inverse Matrices
King Saud University
Definition
• Last time we said the the inverse of an n by
n matrix A is an n by n matrix B where,
AB = BA = In.
• We also talked about how to find the
inverse of a matrix and said that not all
matrices have inverses (some are singular)
so won’t review that here.
Properties of Inverses
1. If A is an invertible matrix then its inverse
is unique.
2. (A-1)-1 = A.
3. (Ak)-1= (A-1)k (we will denote this as A-k )
4. (cA)-1 = (1/c)A-1, c ≠ 0.
5. ( AT)-1 = (A-1)T.
Some theorems involving Inverses
1. If A and B are invertible matrices then,
(AB)-1 = B-1A-1.
2. If C is an invertible matrix then the following
properties hold.
a) If AC = BC then A = B.
b) If CA = CB then A = B.
3. If A is an invertible matrix, then the system of
equations Ax = b has a unique solution given by
x = A-1b.
Elementary Matrices
• An n by n matrix is called an elementary
matrix if it can be obtained from In by a
single elementary row operation.
• These matrices allow us to do row
operations with matrix multiplication.
Representing Elementary Row Operations
Theorem: Let E be the elementary matrix
obtained by performing an elementary row
operation on In. If that same row operation
is performed on an m by n matrix A, then
the resulting matrix is given by the product
EA.
Row equivalent matrices
• Let A and B be m by n matrices. Matrix B is
row equivalent to A if there exists a finite
number of elementary matrices E1, E2, ... Ek
such that
B = EkEk-1 . . . E2E1A.