How do you Transpose a Matrix?

How do you Transpose a Matrix?

Transpose of a matrix

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by AT or A′.
How do you Transpose a Matrix 1From the definition it is obvious that if order of A is m × n, then order of AT is n × m.
Example:
How do you Transpose a Matrix 2

Properties of transpose

Let A and B be two matrices then,

  1. (AT)T = A
  2. (A + B)T = AT + BT, A and B being of the same order
  3. (kA)T = kAT, k be any scalar (real or complex)
  4. (AB)T = BTAT, A and B being conformable for the product AB
  5. (A1 A2 A3 …… An-1 An)T = AnT An-1T ……. A3T A2T A1T
  6. IT = I

Symmetric and Skew-symmetric Matrices

(1) Symmetric matrix :
A square matrix A = [aij] is called symmetric matrix if aij = aji for all i, j or AT = A.
How do you Transpose a Matrix 3
(2) Skew-symmetric matrix :
A square matrix A = [aij] is called skew- symmetric matrix if aij = −aji for all i, j or AT = −A.
How do you Transpose a Matrix 4
All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.
aij = −aji ⇒ aij = 0

Properties of symmetric and skew-symmetric matrices

  1. If A is a square matrix, then A + AT, AAT, ATA are symmetric matrices, while A − AT is skew- symmetric matrix.
  2. If A is a symmetric matrix, then −A, KA, AT, An, A−1, BT AB are also symmetric matrices, where n ∈ N, K ∈ R and B is a square matrix of order that of A.
  3. If A is a skew-symmetric matrix, then
    (a) A2n is a symmetric matrix for n ∈ N.
    (b) A2n+1 is a skew-symmetric matrix for n ∈ N.
    (c) kA is also skew-symmetric matrix, where k ∈ R.
    (d) BT AB is also skew- symmetric matrix where B is a square matrix of order that of A.
  4. If A, B are two symmetric matrices, then
    (a) A ± B, AB + BA are also symmetric matrices,
    (b) AB – BA is a skew- symmetric matrix,
    (c) AB is a symmetric matrix, when AB = BA.
  5. If A, B  are two skew-symmetric matrices, then
    (a) A ± B, AB – BA are skew-symmetric matrices,
    (b) AB + BA is a symmetric matrix.
  6. If A a skew-symmetric matrix and C is a column matrix, then CT AC is a zero matrix.