## Matrices

# Transpose of a Matrix

**Transpose of a Matrix :**The transpose of a matrix A is denoted by A^{T}(or) A' and it is obtained by interchanging the rows into columns and columns into rows.

Ex: \tt A=\begin{bmatrix}a & b \\c & d \end{bmatrix}

\tt A^{T}=\begin{bmatrix}a & c \\b & d \end{bmatrix}

→ [A]_{m × n}⇒ [A^{T}]_{n }_{× m}**Properties of transpose :**→ (A^{T})^{T}= A

→ (A ± B)^{T}= A^{T}± B^{T}

→ (AB)^{T}= B^{T}A^{T}

→ (KA)^{T}= KA^{T}for any scalar K.**Orthogonal matrix :**A square matrix A is said to be orthogonal if A · A^{T}= A^{T}· A = I.- The transpose of orthogonal matrix is also orthogonal .
- If A is orthogonal matrix then |A| = ± 1

### View the Topic in this video From 09:55 To 19:43

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1. **Properties of transpose :** Let A and B be two matrices then,

(i) (*A ^{T}*)

*=*

^{T}*A*

(ii) (*A* + *B*)* ^{T}* =

*A*+

^{T}*B*,

^{T}*A*and

*B*being of the same order

(iii) (*kA*)* ^{T}* =

*kA*,

^{T}*k*be any scalar (real or complex)

(iv) (*AB*)* ^{T}* =

*B*,

^{T}A^{T}*A*and

*B*being conformable for the product

*AB*

(v) (*A*_{1}*A*_{2}*A*_{3} .... *A*_{n−1} *A _{n}*)

*=*

^{T}*A*

_{n}^{T}A_{n−1}

*......*

^{T}*A*

_{3}

^{T }A_{2}

^{T }A_{1}

^{T} (vi) *I ^{T}* =

*I*