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BodhiAI
Nov. 5, 2019

Transpose of a matrix, Orthogonal ,Symmetric Matrix:

TRANSPOSE OF A MATRIX
Let A= [aij] m×n is a matrix oforder m×n, then the transpose of A can be obtained by changing all
rows to columns and allcolumns to rows, i.e.
Transpose of A= [aji]n×m
Usually, the transpose of A is denoted by AT or A'.
PROPERTIES OF TRANSPOSE
Let A= [aij]m×n is a matrix oforder m×n, then the transpose of A can be obtained by changing all
rows to columns and allcolumns to rows, i.e.
P-1 If A and B are any two matrices of same order, then

(A ± B)T= AT±BT

P-2 If A= [aij] is any matrix of order m×n and λ be any scalar, then

(kA)T=(k[aij])T
 (kA)T = [kaij]T= k(A)T
P-3 Transpose of Transpose of matrix A is the matrix itself
i.e. (AT)T = A
P-4 REVERSAL LAW
Transpose of product is the product of transposes taken in reverse order.
If A and B are any two matricesconformable for multiplication, then
(AB)T = BTAT
Similarly, IfA, B, C are any three matrices conformable for multiplication,
then, (ABC) T =CTBTAT and similar other cases.