### 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 = a_{11} + a_{22} + —-+ a_{nn}

### Cofactor Matrix

If A is square matrix (a_{ij}) and A_{11}, A_{12} — etc. are the Cofactor of elements a_{11}, a_{12} — 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^{-1}A = AA^{-1} = 1

• (AB)^{-1} = B^{-1}A^{-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.