# Transpose of a Matrix

• Transpose of a Matrix : The transpose of a matrix A is denoted by AT (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 ⇒ [AT]n × m
• Properties of transpose :
→ (AT)T = A
→ (A ± B)T = AT ± BT
→ (AB)T = BT AT
→ (KA)T = KAT for any scalar K.
• Orthogonal matrix : A square matrix A is said to be orthogonal if A · AT = AT · 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) (AT)T = A

(ii) (A + B)T = AT + BT, A and B being of the same order

(iii) (kA)T = kAT, k be any scalar (real or complex)

(iv) (AB)T = BTAT, A and B being conformable for the product AB

(v) (A1A2A3 .... An−1 An)T = AnT An−1T ...... A3T A2T A1T

(vi) IT = I