# TYPES OF MATRICES

#### Definition: Matrix:

• A set of mn elements arranged in a rectangular arrangement along m rows and n columns enclosed by the brackets [ ] or ( ) is called m by n matrix.
• Example:

#### Order of matrix

• The order of a matrix is a number of rows and columns of the matrix.
• Example:
• From above example the order of matrix ot has no of rows=2 and no of columns=4.

#### Row matrix

• The Matrix has a single row is called a row matrix.
• Example:

#### Column matrix

• The matrix has a single column is called a column matrix.
• Example:

#### Square matrix

• A matrix having an equal number of rows and columns is called a square matrix.
• Example:

#### Null matrix

• In a matrix, if all the elements are zero then that matrix is called a null or zero matrices and it is always denoted by 0.
• Example:

#### Diagonal matrix

• In a square matrix, all the elements except the elements in the main diagonal are zeros, then the matrix is called a diagonal matrix.
• Example:

#### Scalar matrix

• A square matrix in which all the elements of its leading diagonal are equal and the other elements all the zero is called a scalar matrix.
• Example

#### Unit matrix

• A diagonal matrix that has unity in the leading diagonal is called a unit matrix. It is always denoted by I.
• Example:

#### Upper triangular matrix

• A square matrix in which all elements below the leading diagonal zero is called an upper triangular matrix.
• Example:

#### Lower triangular matrix

• A square matrix in which all elements above the leading diagonal are zero is called a lower triangular matrix.
• Example:

Transpose of a matrix

• The matrix got from the given matrix A, by interchanging the rows and columns is called the transpose of A and denoted by $A^T$.
• Example:

#### Symmetric matrix

• A square matrix A is symmetric if  $A&space;=A^T$.
• Example:

#### Skew symmetric matrix

• In skew- symmetric A square matrix, A then skew-symmetric is $A=-A^T$.
• Example:

#### Singular matrix

• In a singular matrix, a square matrix A is said to be singular if the value of A is zero.
• Example:

#### Equality of matrices

• Two matrices A and B is said to be equal if and only if
1.  Both matrices of the same order
2.  Each element A is equal to the corresponding element of B.
3. Example:

• Let A and B be two matrices of same orders, then their sum A+B is defined as the matrix where each element is the sum of the corresponding elements of A and B.
• Example:

#### Subtraction of matrices

• Let A and B be two matrices of the same orders, then their difference A-B is defined as the matrix where each element is the difference of the corresponding elements of A and B.
• Example:

#### Scalar Multiplication

• The product of a scalar matrix A by a scalar K is a matrix in which each element is K times the corresponding elements of A.
• Example:  k= 2, $A=\begin{bmatrix}&space;1&space;&4&space;&2&space;\\&space;7&space;&&space;0&space;&1&space;\\&space;2&3&space;&-1&space;\end{bmatrix}\Rightarrow&space;KA=2\begin{bmatrix}&space;1&space;&4&space;&2&space;\\&space;7&&space;0&space;&&space;1\\&space;2&&space;3&space;&-1&space;\end{bmatrix}\Rightarrow&space;\begin{bmatrix}&space;1\times&space;2&space;&&space;4\times&space;2&space;&2\times&space;2&space;\\&space;7\times&space;2&&space;0\times&space;2&space;&1\times&space;2&space;\\&space;2\times&space;2&&space;3\times&space;2&-1\times&space;2&space;\end{bmatrix}\Rightarrow&space;\begin{bmatrix}&space;2&space;&8&space;&4&space;\\&space;14&space;&0&space;&&space;2\\&space;4&space;&&space;6&space;&-2&space;\end{bmatrix}$

#### Multiplication of matrices

• Two matrices can be multiplied only when the number of columns in the first matrix is equal to the number of rows in the second matrix.
• E.g., if A = m×n, B = n×p then AB = m×p
• Example: $A=\begin{bmatrix}&space;3&space;&&space;1\\&space;6&space;&&space;4&space;\end{bmatrix};B=\begin{bmatrix}&space;2&&space;7&space;&10&space;\\&space;5&space;&&space;8&space;&&space;9&space;\end{bmatrix}\Rightarrow&space;A\times&space;B=\begin{bmatrix}&space;3\times&space;2+1\times&space;5&space;&3\times&space;7+1\times&space;8&space;&3\times&space;10+1\times&space;9&space;\\&space;6\times&space;2+4\times&space;5&space;&6\times&space;7+4\times&space;8&space;&&space;6\times&space;10+4\times&space;9&space;\end{bmatrix}\Rightarrow&space;\begin{bmatrix}&space;6+5&space;&21+8&space;&30+9&space;\\&space;12+20&space;&42+32&space;&&space;60+36&space;\end{bmatrix}\Rightarrow&space;\begin{bmatrix}&space;11&space;&29&space;&39&space;\\&space;32&74&space;&96&space;\end{bmatrix}$