BCA-203: Discrete Mathematics

Unit 1 - Matrix

Type of matrix

1.       Matrix

2.       Principle Matrix

3.       Transpose Matrix

4.       Equal Matrix

Note: M = Row, N = Column

-Matrix and Corresponding Elements are same is called Equal Matrix.

-Convert in to Row and Column is called Transpose Matrix.

Column Matrix:

                If any matrix has order M×1 than it is known as column matrix i.e. If a matrix has only one column and any number of rows than it is known as Column matrix.

Example:

Row Matrix:

                If a matrix has order 1×N than it is known as row matrix that is if a matrix has single row and any number of columns than it is known as row matrix.

Example:             A= [a₁₁  a₁₂           a₁₃          a₁₄           - - -         a_n]

Square Matrix:

                If a matrix has number of rows and number of column are equal than it is as square matrix. For square matrix N=N there for order of square matrix is denoted by am also.

                A= [a_ij]_n

                B= [b_ij]₃

Null Matrix:

                A all the elements of a matrix are 0 than it is called null matrix or 0 matrix.

Simple Definition: All elements are 0 is called Null matrix.

Example:

Unique Matrix & Identity Matrix:

                In a square matrix if all the principle diagonal elements are 1 and all others elements are 0 than it is called unique matrix & identity matrix

                Matrix a order of N×N it is a unique matrix when it is denoted by

Simple Definition: All the principle diagonal matrix are 1 and other elements are 0 is called unique matrix & Identity Matrix

Example:

Diagonal Matrix:

                Ina a square matrix, all the elements are 0 except the principle diagonal elements is called as diagonal matrix.

Simple Definition: All the elements are 0 and Principle diagonal elements are other is called Diagonal matrix.

Example:

Scalar Matrix:

                In a square matrix all the principle diagonal elements are same and all other elements are 0 then it is called scalar matrix.

Simple Definition: All the principle diagonal elements are same and other elements are 0 is called scalar matrix.

Example:

Symmetric Matrix:

                For any square matrix a, transpose of matrix a is equal to the given matrix a i.e. A^t=A, than it is called symmetric matrix.

In other words a_ij=a_ji for A= [a_ij]_n , when

                I= 1, 2, -----n

                J= 1, 2, -----n

Simple Definition: Rows and Column are equal is called symmetric matrix.

Example:

                A≠A^T

Example:

A=A^T

a₁₂ = a₂₁

a₁₃ = a₃₁

a₂₃ = a₃₂

Skew Symmetric Matrix:

                For any square matrix a, a_ij = -a_ji for Ɏ_ij or A_T = A, and all the principle diagonal elements are 0 it is called skew symmetric matrix.

Simple Definition: Principle diagonal elements are 0 and other elements are equal is called skew symmetric matrix.

Example:

a_ij = a_ji

a₁₂ = a₂₁

a₁₃ = a₃₁

a₂₃ = a₃₂ 

A^T = -A

Triangle Matrix:

1)      Upper Triangle Matrix:

For a square matrix if each element below the principle diagonal is 0 than it is called has upper triangle matrix.

Example:

2)      Lower Triangle Matrix:

For a square matrix if each element above the principle diagonal is 0 than it is called has lower tringle matrix.

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