Matrices, like any other expressions,are also subjected to basic algebraic operations. Multiplication is one of those. In addition to normal restrictions on matrix operations, multiplication of two matrices has additional restrictions and the method is a bit strange.
Suppose A and B are two matrices, their product is defined only if the number of columns of A and the number of rows of B must be same. Else, the multiplication is not defined. The algorithm for multiplying 3x3 matrices is described below.
The first element of first row and first column of the product C of matrix A and matrix B is the sum of the products of the elements of the first row with the corresponding elements of the first column of B.
For the second element in the same row, the products are now first row of Avs second column of B. Like this the entire elements of C are found.
With the restrictions and strange algorithm of the process, it can clearly be concluded that multiplication of matrices is not commutative.
A matrix just conveys the details of the data and it does not give any solution to any question on the relations between the items. On the other hand if the same arrangement of data indicates an operation for a solution, then that particular arrangement is called the determinant of the same matrix.
A matrix is denoted by containing the items of a data within a set of elongated square brackets. In a determinant the same are enclosed by two thin vertical bars (similar to absolute value symbol). The following is an example.
The determinant of the matrix indicates the solution by the operation (a1 *b2- a2*b1).
The determinant of a matrix with 3 rows and 3 columns is called a determinant 3x3 matrix or just as determinant 3x3.
To find determinant of 3x3 matrix, first rewrite the elements in the same order but with the symbol of a determinant. To find the value of determinant of a 3x3, the following is the procedure.
Consider a11, the first element in the first row and column. By deleting the row and column that includes a11, you are left with a 2 x 2 determinant which is called as the minor of the element a11and denoted as, M11.
Same way, for the second and third elements in the first row a12 and a13 , the minors are M12 and M13 respectively. Now the formula for the value of the determinant D is,
D = (a11*M11 -a12*M12 + a13*M13)