To compute the product of two matrices, we can follow a process where each element in the first row of the first matrix is multiplied by the corresponding element in the first column of the second matrix. These products are then added together. This process continues until each row of the first matrix is multiplied by each column of the second matrix.
In the provided example, the computation of the resulting matrix (prod) involves multiplying specific rows of the first matrix with corresponding columns of the second matrix. For instance, to determine the value of the element prod[0,0], we multiply the first row of the first matrix (1, 3, 2) with the first column of the second matrix (2, 1, 1). Subsequently, we sum the products of the corresponding elements, resulting in (12) + (31) + (2*1) = 7.
Two matrices can be multiplied only if the number of columns in the first matrix is equal to the number of rows in the second matrix. If the dimensions of matrix A are p × q and matrix B are q × r, then the resulting matrix will have dimensions of p × r. Matrix Product can be represented as:
C<sub>ij</sub> = Σ A<sub>ik</sub>B<sub>kj</sub>
Algorithm to Find the Product of Two Matrices
-
- STEP 1: START
- STEP 2: DEFINE row1, col1, row2, col2
- STEP 3: INITIALIZE matrix a[][] ={{1,3,2},{3,1,1}, {1,2,2}}
- STEP 4: INITIALIZE matrix b[][] ={{2,1,1},{1,0,1}, {1,3,1}}
- STEP 5: row1 = a.length
- STEP 6: col1 = a[0].length
- STEP 7: row2 =b.length
- STEP 8: row2 = b[0].length
- STEP 9: if(col1!=row2)
PRINT “Matrices cannot be multiplied”
else
Go to step 10; - STEP 10: prod[][] = [row1][col2]
- STEP 11: REPEAT STEP 12 to STEP 14 UNTIL i<row1
//for(i=0; i<row1; i++) - STEP 12: REPEAT STEP 13 to STEP 14 UNTIL j<col2 // for(j=0; j<col2; j++)
If(j>i) then PRINT 0 else PRINT a[i][j] - STEP 13: REPEAT STEP 14 UNTIL k<row2 // for(k=0; k<row2; k++)
- STEP 14: prod[i][j] = prod[i][j] + a[i][k]*b[k][j]
- STEP 15: REPEAT STEP 16 to STEP 18 UNTIL i<row1
- STEP 16: REPEAT STEP 17 UNTIL j<col2
- STEP 17: PRINT prod[i][j]
- STEP 18: PRINT new line
- STEP 19: END
Java Program to Find the Product of Two Matrices
public class ProdMatrix
{
public static void main(String[] args) {
int row1, col1, row2, col2;
//Initialize matrix a
int a[][] = {
{1, 3, 2},
{3, 1, 1},
{1, 2, 2}
};
//Initialize matrix b
int b[][] = {
{2, 1, 1},
{1, 0, 1},
{1, 3, 1}
};
//Calculates number of rows and columns present in first matrix
row1 = a.length;
col1 = a[0].length;
//Calculates the number of rows and columns present in the second matrix
row2 = b.length;
col2 = b[0].length;
//For two matrices to be multiplied,
//number of columns in first matrix must be equal to number of rows in second matrix
if(col1 != row2){
System.out.println("Matrices cannot be multiplied");
}
else{
//Array prod will hold the result
int prod[][] = new int[row1][col2];
//Performs product of matrices a and b. Store the result in matrix prod
for(int i = 0; i < row1; i++){
for(int j = 0; j < col2; j++){
for(int k = 0; k < row2; k++){
prod[i][j] = prod[i][j] + a[i][k] * b[k][j];
}
}
}
System.out.println("Product of two matrices: ");
for(int i = 0; i < row1; i++){
for(int j = 0; j < col2; j++){
System.out.print(prod[i][j] + " ");
}
System.out.println();
}
}
}
}
Output:
Product of two matrices 7 7 6 8 6 5 6 7 5
In conclusion, the provided Java program allows for the computation of the Product of two matrices. The program verifies the condition that the number of columns in the first matrix must be equal to the number of rows in the second matrix. Follow tutorials.freshersnow.com to learn more.