5. The Transpose Of A Matrix : (Changing rows & columns)
Let A be any matrix . Then, A = [ aij ] of order m × n
Þ AT or = [aji] for 1 ć i £ n & 1 £ j £ m of order n × m
Thus AT is obtained by changing its rows into column and columns into row.
Properties of Transpose :
If AT & BT denote the transpose of A and B ,
(a) (AT)T = A
(b) (A + B)T = AT + BT note that A & B have the same order.
Proof : [(A + B)T]i j = [(A + B) ]ji
= a j i + bj i = (AT)i j + (BT)i j = (AT + BT )i j
(c) (KA)T = KAT , K be any scalar (real or complex )
(d) (A B)T = BT AT A & B are conformable for matrix product AB
Proof : A = [ai j] is m × n ; B = [bi j ] is n × p
[(AB)T]i j = (AB)j i
= r=1ån aij b ri = r=1ån (AT)rj(BT) ir
=r=1ån(BT)ir (AT)rj = (BT AT )i j
General : (A1,A2,.................An)T = AnT,..............,A1t (reversal law for transpose)
Example 1. A=
B =
verify (AB)T = BT AT
Example 2. A=
B =
verify (A + B)T = BT + AT.
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