FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.61Properties of Matrix Multiplication(i) A(BC) = (AB) C (Associative law) (Where Amxn Bnxs & Csxt)
(ii) A(B+C) = AB + AC (Distributive law)
(iii) AI = IA = A
(where A is square matrix and I is unit matrix of same order)
(iv) A x O = O x A = O
(where A is matrix of order m x n & O is matrix of order n x m)
(v) In General AB ≠ BA (Not Commutative)
(Note AB = BA only where B is equal to Adjoint of A)
(vi) IF AB = 0, doesn’t imply that
A = 0 or B = 0 or both = 0
SELF EXAMINATION QUESTIONS
(1) Evaluate AB
where A= 0 10 2 & B= 0 03 4 Ans. 0 00 0
(2) Show that AI = IA = A
where A^29
6 5=
−
(3) Evaluate AB & BA. Is AB = BA?
where A= ^3425 B= 82 79 (Ans. No.)
TRANSPOSE OF A MATRIX
Transpose of matrix A denoted by A’ or At. Transpose of A can be obtained by inter changing rows and
columns of a matrix A.
If A = [aij]mxn then A’ = [aij]nxm
Thus, If A
2 x 3
3 2 4
7 8 9
=
Then A’
3 x 2
3 7
2 8
4 9
=
Properties of Transpose of a matrix
(i) (A)’ = A
(ii) (A + B)’ = A’ + B’
(iii) (kA)’ = kA’ (Where k is a Scalar)
(iv) (AB)’ = B’A’