Transpose of a Matrix

If A = [aij] is mxn matrix, then the matrix of order n x m obtained by interchanging the rows and columns of A is called the transpose of A.It is denoted At or A, for example:

Properties of Transpose of a Matrix

• If A and B are tranpose of A and B respectively, then
• (A) = A
• (A + B) = A + B, A and B being comparable.
• (KA) = KA, K being any complex number.
• If A and B are of suitable size for AB to exist, then (A B) = B + A (Reversal Law of Transposes)

Symmetric Matrix

A square matrix A is called symmetric if A = At, for example:

Thus A is symmetric

Skew Symmetric

A square matrix A is called skew symmetric if A = -At, for example:

Singular Matrices

A square matrix A is called singular if |A| = 0 , for example if

Non-singular Matrices

A square matrix A is called singular if |A| != 0 , for example if

Trace of Matrix

If A is square nxn matrix, its trace, denoted by tr A, is defined to be the sum of the term on the leading diagonal.

tr A = a11 + a22 + —-+ ann

Cofactor Matrix

If A is square matrix (aij) and A11, A12 — etc. are the Cofactor of elements a11, a12 — etc. respectively, then matrix [Aij] is called cofactor Matrix of A.

The transpose of cofactor matrix of a given matrix A is called the Adjoint of A or adj A.
Let A = (aij) be a square matrix of order n x n and (cij) is a matrix obtained by replacing each element aij by its corresponding cofactor cij then (cij)t is called the adjoint of A.

Inverse of a Matrix

If A and B are two square matrices such that AB = BA = 1, then B is called the Inverse of A and vice-versa.
A (adj A)|A| = I, Its behaves as the inverse of A. Hence

If A is a non-singular square matrix , then

The necessary and sufficient condition for a square matrix to have an inverse is that it should be a non-singular matrix.

Properties

• A-1A = AA-1 = 1
• (AB)-1 = B-1A-1
• (A-1)-1 = A