Mathematics TRANSPOSE OF A MATRIX

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TRANSPOSE OF A MATRIX

Transpose of a Matrix

If `A = [a_(ij)]` be an `m - n` matrix, then the matrix obtained by interchanging the rows and columns of `A` is called the transpose of `A`. Transpose of the matrix `A` is denoted by `A′` or `(A^T)`. In other words, if `A = [a_(ij)]_(m - n)`, then `A^T = [a_(ji)] n - m`. For example

if `A = [(3,5 ),(sqrt3, 1),(0, -1/5)]_(3xx2) ,` the `A^T = [ (3,sqrt3, 0),(5, 1, -1/5)]_(2xx3)`

Properties of transpose of the matrices

We now state the following properties of transpose of matrices without proof. These may be verified by taking suitable examples.

For any matrices A and B of suitable orders, we have

(i) `(A^T)^T = A`

(ii) `(kA)^T = kA^T` (where `k` is any constant)

(iii) `(A + B)^T =A^T +B^T`

(iv) `(AB)^T= B^TA^T`

`text(Note :)`

`I^T= I,` where I is an identity matrix.