Every square matrix is associated with a real number which is called the determinant of the square matrix. Let S be the set of all square matrices and R be the set of real numbers. Then the function f:S->R is called the determinant. The determinant of a matrix M is denoted by |M| or detM. It can be computed using a specific mathematical operation on the elements of the matrix.
The determinant of a matrix is used to solve the system of equations. Also, the determinant of a matrix gives an important information when the elements of a matrix are associated with a linear transformation.
Determinant of a 2×2 matrix:
The determinant of a 2×2 matrix 
is defined as ad-bc. It is nothing but the difference of cross products of the elements in order.
Example:
The determinant of 
is 2*2 –(3*-1)=4+3=7.
Minor and Co-factor of an element:
Minor of an element of a 3×3 matrix is the determinant of 2×2 matrix obtained after leaving the row and column containing it.
The cofactor of an element is the product of its minor with (-1)i+j where i is the row number and j is the column number containing it.
Example:
Let A =
Minor of 1 =
=-5+8=3; Co-factor of 1 = (-1)1+1 3 = 3
Minor of 4 =
=-2-4 =-6.; Co-factor of 4 = (-1)1+2 (-6) =(-1)(-6)=6
Determinant of a 3×3 matrix:
The determinant of a 3×3 matrix is the sum of the products of elements of any row (or column) with their corresponding cofactors.
Example:
Let A =
Its corresponding cofactor matrix is 
So, if we use the first row to find the determinant, taking the sum of the products of first row elements with the corresponding elements of the co-factor matrix then,
|A| = 1*3 + 4*6 + (-2)*9 = 3 +24-18 = 9
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