Here you will learn what is transpose of matrix with definition and examples.
Let’s begin –
What is Transpose of Matrix
Let A = [aij] be a m×n matrix. Then the transpose of A, denoted by AT or A’, is an n×m matrix such that
(AT)ij = aji for all i = 1, 2, ….. , m; j = 1, 2, ….., n.
Thus, AT is obtained from A by changing its rows into columns and columns into rows.
for example, if A = [123423413214],
then AT = [123232341414]
The first row of AT is the first column of A. The second row of AT is the second column of A and so on.
Properties of Transpose
(a) for any matrix A, (AT)T = A.
(b) for any two matrices A and B of the same order, (A+B)T) = AT + BT.
(c) If A is a matrix and k is a scalar, then (kA)T = k(AT).
(d) If A and B are two matrices such that AB is defined, then (AB)T = BTAT.
Generalisation : If A, B, C are three matrices confirmable for the products (AB)C and A(BC), then (ABC)T = CTBTAT.
The above law is called reversal law for transposes i.e. the transpose of the product is the product of the transposes taken in the reverse order.
Example : If A = [−123] and B = [−2−1−4], verify that (AB)T = BT AT
Solution : We have,
A = [−123] and B = [−2−1−4]
∴ AB = [−123] [−2−1−4] = [214−4−2−8−6−3−12]
⟹ (AB)T = [2−4−61−2−34−8−12]
Also, BT = [−2−1−4] and AT = [−123]
BTAT = [−2−1−4] [−123] = [214−4−2−8−6−3−12]
Hence (AB)T = BT AT