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 + ......... + a1nxn = b1
a21x1 + a22x2 + ......... + a2nxn = b2
am1xa + am2x2 +.........+ amnxn = bm
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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