Matrices & Determinants - 2

Null Matrix :-

MAtrix with all elements O.

Transpose of a Matrix:- The matrix obtaired from a given matrix A, by inter changing rows and columns is called transpose of A and is denoted by a^T . or A ¹.

Properties of Transpose (Secondary In formation):-

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

(ii) (A + B)^T = A^T + B^T

(iii) (A)^T = . A^T

(iv) (A B)^T = B^T A^T.

Conjugate of a Matrix:- A matrix obtained from any given matrix A coutaining complex number as its elements, or replaing its elements by the corresponding conjugate complex no is called conjugate of A and is denoted by .

Properties of conjugate (Secondary Information:-

(i) = A

(ii)

(iii)

(iv)

Transpose Conjugate of a Matrix:-

The transpose of conjugate of a matrix denoted by .

Properties of Transpose of conjugate:-

(Secondary Information):-

(i)

(ii)

(iii)

(iv)

ALGEBRA OF MATRICES:-

Addition and Subtraction:- Any two matrices can be added lif they are of same order and the (or Subtrected)

resulting matrix is of same order, corresponding elements are added or subtracted .

Scalar Multiplication:-

The matrix obtained by multiplleying every element of by a scalar is called the scalar multiple of A by and is denoted by A .

Multipliecation of Matrices:-

Two matrices can be multiplied lonly when the no of columns in the first is equal to the no . of rows inm the second. Such matrices are called conformable for multiplication.

If A B = c
A - [a_ij] _mXn B - [b_ke] _nXp

C _ij² a _ik b _kj

Special Matrices (Secondary Information:-

Symmetric and Skew Symmetric Matrices:-

A square matriex is A is said to be symmetric if
A = A^T.

and skew symmetric if A = - A^T.

Unitary Matrix

A matrix is unitary if A = 1