mathematics - Determinants and Matrices, Math

Determinant: Consider the set of linear equations a₁x + b₁y = 0 and a₂x + b₂y = 0, where on eliminating x and y we get the eliminant a₁b₂ - a₂b₁ = 0; or symbolically, we write in the determinant 

 

2×2 order having 2 rows and 2 columns. Similarly, a determinant of 3 × 3 order can be expanded as:

 

 

The sign is determined as positive or negative according as the permutation is even or odd. 

 

Properties of a Determinant: 

1. If all the elements of a row (column) are zero, then the value of the determinant is zero. 

2. If the rows (columns) of a determinant are changed into columns (rows) the value of the determinant remains unaltered. 

3. If the elements of a row (column) are identical proportional to the elements of any other row (column), then the determinant vanishes. 

4. The interchange of any two rows (columns) of a determinant changes its value in its sign only. 

5. If all the elements of a row (column) of a determinant are multiplied by a constant k, then the value of the determinant gets multiplied by k. 

6. If the elements of a row (column) of a determinant area expressed as the sum (difference) of two quantities, then the determinant can be expressed as the sum (difference) of two determinants of the same order.

Matrices: If a system of m linear equations in n unknowns is given as

a₁₁X₁ + a₁₂x₂ + ......... + a_1nx_n = b₁

a₂₁x₁ + a₂₂x₂ + ......... + a_2nx_n = b₂

................................................

a_m1xa + a_m2x₂ +.........+ a_mnx_n = b_m

 

Called an m × n .matrix with elements a1(i = 1, 2, ..... m; J = 1. 2.. n) over the field of real (complex) numbers. 

Types of Matrices 

Square matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix. 

Identity matrix: A square matrix A all of whose non-diagonal elements are zero (i.e., it is a diagonal matrix) and also all the diagonal elements are unity is called a unit matrix or an identity matrix. 

Zero matrix or Null matrix: Any m × n matrix in which all the elements are zero is called a zero matrix or null matrix of the type m × n and is denoted by Om×n.

Row matrix: A 1 × n matrix having only one row is called a row matrix. e.g., A = [a11 a12 ....... a1n]1xn. 

Column matrix: A n × 1 matrix having only one column is called a column matrix 

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