Properties Of Matrix Multiplication :
1. Matrix multiplication is not commutative.
A =
B =
AB =
BA =
Þ AB ¹ BA (in general)
Note :
(i) If A and B are two non-zero matrices such that AB =0 then A and B are called the divisors of zero. Also AB = 0 ¹ BA = 0
Example : AB=
Þ AB = 0 ¹ A = 0 or B = 0
If A and B are two matrices such that
(i) AB = BA Þ A and B commute each other
[A2 – B2 = (A–B) (A+B) if A and B commute ]
(ii) AB = –BA Þ A and B anti commute each other
2. Matrix Multiplication Is Associative :
If A, B & C are conformable for the product AB & BC, then
(A . B) . C = A. (B.C)
A = [aij] is m × n ; B = [bij] is n × p ; C = [cij] is p × q
Note : (A . B) . C & A . (B . C) have the same order Þ comparable.
[(A B) . C ]ij = r=1åp (AB)ir Crj
(associativity in R)
= [ A . (B C) ]i j
[(A .B) . C]i j = [A .(B . C)]i j Þ (A B) C = A . ( B C )
Example1: If A, B, C are the given matries such that AB = 0 and BC = I then
prove that (A + B)2 (A + C)2 = I where I is an identity matrix.
[Solution : Given BC = I and AB = 0
or A (BC) = AI = A Þ (AB) (C) = A Þ A = 0 ;
hence (A + B)2 (A+C)2 = B2 C2 = (BB) (CC) = B (BC)(C) = B I C = BC = I ]
3. Distributivity :
Provided A,B & C are conformable for respective products
[A . (B + C ]i j = r=1ån air (B+C)r j = r=1å n ar j (br j +cr j)
=r=1ån ai r br j + r=1ån ai r +cr j
= (A B)i j + (A C)i j = (AB + AC)i j
4. Positive Integral Powers of A square matrix :
for a square matrix A, A2A = (AA) A = A (AA)=A3.
Note that for a unit matrix I of any order , Im = I for all m Î N.
It can be easily seen that Am . An = Am+n and (Am)n = Amn
Matrix Polynomial :
(a) Idempotent Matrix : A square matrix is idempotent provide A2 = A.
Note that An = A " n ³ 2, n Î N .
(b) Periodic Matrix : A square matrix is which satisfies the relation Ak+1 = A, for some positive integer K, is a periodic matrix. The period of the matrix is the least value of K for which this holds true.e.g. the matrix
has the period 1.
Note that period of an idempotent matrix is 1.
(c) Nilpotent Matrix : A square matrix is said to be nilpotent matrix of order m, (m Î N) , if
Am = 0, Am–1 ¹ 0.
Note that a nilpotent matrix will not be invertible.
(d) Involutary Matrix : If A2 = I, the matix is said to be an involutary matirx.
Note that A = A –1 for an involutary matrix.
(e) If A be a2 square matrix of order n then f (A) = a0An + a1An–1 + a2An–2 +..........+ anIn is called a matrix polynomial.
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