BodhiAI

Nov. 5, 2019
**TRANSPOSE OF A MATRIX**

Let A= [a_{ij}] 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= [a_{ji}]_{n×m}

Usually, the transpose of A is denoted by A^{T} or A'.**PROPERTIES OF TRANSPOSE**

Let A= [a_{ij}]_{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}= A^{T}±B^{T}

**P-2** If A= [a_{ij}] is any matrix of order m×n and λ be any scalar, then

(kA)^{T}=(k[a_{ij}])^{T}

(kA)^{T} = [ka_{ij}]^{T}= k(A)^{T}**P-3** Transpose of Transpose of matrix A is the matrix itself

i.e. (A^{T})^{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} = B^{T}A^{T}

Similarly, IfA, B, C are any three matrices conformable for multiplication,

then, (ABC) ^{T} =C^{T}B^{T}A^{T} and similar other cases.