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matrix competencies

Do these as an applet using a class basic matrix operations

1. Write a program that randomly fills a 3 by 4 by 6 array, then prints the largest and smallest values in the array.

2.  

• Matrices are two-dimensional arrays of numbers. You can think of a matrix as a sort of table of numbers with rows and columns

  3	4	9.8
  17	5.55	3.14159
  2.71828	1	0
  -5	8	4

  Array elements are identified by row and column indexes. The table above has 4 rows 3 columns. each element in the table can be accessed by a row and column index. For example the element -5 is at row 3 and column 0.

• The lab is designed to give you experience with writing the matrix operations of addition and multiplication using two dimensional arrays in Java.
• The lab has two parts that modify the Matrix class:
  1. Write a method to add two matrices for the class Matrix.
  2. Add another method to the Matrix class to multiply two arrays.
  3. Add the printMatrix() method to Matrix class.

Matrix Addition

• Matrix addition involves adding up the corresponding elements in two matrices to form their sum.
  • For addition, matrices must be of the same size (the same number of rows and columns. This is a condition that must be checked for.
  • Addition than occurs by adding corresponding elements of the two matrices together. For example :
    • If three matrices A, B, C have the same number of rows and columns
    • Then C(3,1) = A(3,1) + B(3,1)
    • and C(0,2) = A(0,2) + B((0,2)
  • A example is Shown in the following

    1 2 7 10 3 4 8 11 5 6 9 12	+	x u r i y v s j z w t k	=	1 + x 2 + u 7 + r 10 + i 3 + y 4 + v 8 + s 11 + j 5 + z 6 + w 9 + t 12 + k
    [A]	+	[B]	=	[C]

•  

  Matrix Multiplication

  • Procedure for matrix multiplication
    • The matrix which is the final product has the same number of rows as the first matrix in the product, and the same number of columns as the second matrix in the product.
    • The number of columns in the first matrix has to be equal to the number of rows in the second matrix.
    • Each element in the product matrix is computed with the following process:
      • Assume element is in row m and column n.
      • Take all of the elements in the mth row of the first matrix, and all of the elements in the nth column of the second matrix.
        • Take the first element element from the chosen row and multiply it by the first element from the chosen column.
        • Add to this the product of the second elements from the row and column.
        • Then add the product of the row's and column's third elements and so forth.
      • The final sum is the value for the (m, n) element in the final product matrix.
  • For Example to Multiply the following the Matrix A * Matrix B = Matrix C

    1 2 7 3 4 8 5 6 9	x	x u r y v s z w t	=	1*x + 2*y + 7*z 1*u + 2*v + 7*w 1*r + 2*s + 7*t 3*x + 4*y + 8*z 3*u + 4*v + 8*w 3*r + 4*s + 8*t 5*x + 6*y + 9*z 5*u + 6*v + 9*w 5*r + 6*s + 9*t
    [A]	x	[B]	=	[C]

3.

Magic squares are squares that contain numbers arranged in such a way that all the rows columns and the two diagonals add up to the same number.

Here is an example of a magic square of size 3:

and an example of a magic square of size 5: