Matrix Part-2

Transpose of a Matrix

If A = [a_ij] 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 A^t 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 = A^t, for example:

Thus A is symmetric

Skew Symmetric

A square matrix A is called skew symmetric if A = -A^t, 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 + —-+ a_nn

Cofactor Matrix

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

Adjoint of a Matrix

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

adj A = (c_ij)^t

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

A^-1 = (adj A)|A| = (adj A)det A

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

I hope, this article will help you a lot to understand the Matrix Part-2. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.