12.2 Solutions to Addition Problems

1. 1.  
      

   2.  
      

   3.  
      

   4. F + B = not possible since the dimensions do not match.
      

   5.  
      

   6.  
      

   7.  
      

   8. A - D = not possible since the dimensions do not match.
      

   9.  
      

   10.  
       

2.  
   

3. 1.  
      

   2. Row three tells us that, in two years, the bears won 11 games and lost 14 games.
      

   3.  
      

4. 1.  
      

   2. Home won 85 to 84.
      

5. Yes. This can be proven using the facts learned in the Questions and Answers in Chapters 1 and 2. The matrix (A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T.
   

6. Yes. The matrices A and B must both be square because they are symmetric. Therefore, if they have the same dimensions, then A - B must also be square. A generic element of A - B is a_ij - b_ij. Since A and B are symmetric, this element is the same as a_ji - b_ji which is also a generic element for (A - B)^T because (A - B)^T = A^T - B^T. Therefore, A - B is symmetric.
   

Computer Program

Qbasic:

REM This program adds and subtracts matrices

REM It uses no commands that are specific to matrices

CLS

PRINT “This program will add or subtract matrices. ”

INPUT “Enter the dimensions of the matrices separated by a comma.”,m,n

DIM a(m,n)

DIM b(m,n)



REM This loop reads the first matrix

PRINT “Please press enter after each element of the matrix. ”

PRINT “Enter all the elements of one row before the next row. ”

FOR i=1 to m

   FOR j=1 to n

      INPUT a(i,j)

   NEXT j

NEXT i



100 REM This allows the user to choose what to do next

INPUT “Do you wish to 1) add 2) subtract or 3) end ”,s

IF s=3 then goto 500



200 PRINT “Enter the next matrix. ”

FOR i=1 to m

   FOR j=1 to n

      INPUT b(i,j)

   NEXT j

NEXT i

If s=1 then GOSUB 1000 ELSE IF s=2 THEN GOSUB 2000 ELSE IF s=3 

             THEN GOTO 500 ELSE GOTO 100

REM   The previous line should be part of the line above it

PRINT “The resulting matrix is: ”

FOR i=1 to m

   FOR j=1 to n

      PRINT a(i,j),

   NEXT j

   PRINT

NEXT i

PRINT

GOTO 100

500 PRINT “Your final solution is: ”

FOR i=1 to m

   FOR j=1 to n

      PRINT a(i,j),

   NEXT j

   PRINT

NEXT i

END



1000 REM This adds matrices

FOR i=1 to m

   FOR i=1 to n

      a(i,j)=a(i,j)+b(i,j)

   NEXT j

NEXT i

RETURN



2000 REM This subtracts matrices

FOR i=1 to m

   FOR i=1 to n

      a(i,j)=a(i,j)-b(i,j)

   NEXT j

NEXT i

RETURN

Pascal:

Program add(input, output);

{This program adds and subtracts matrices.}



uses crt; {Necessary for some Pascal compilers}



type

   matrix=array[1..10,1..10] of real;



var

   choice : integer;  {for choosing operation}

   m,n: integer; {dimensions of the matrices}

   a,b: matrix;  {matrix}



procedure readmatrix(var a:matrix; m,n:integer);

   var 

      i,j: integer; {counters}



   begin {read}

   for i:=1 to m do

      begin {do}

      for j:=1 to n do 

         read(a[i,j]);

         readln;

      end {do}

   end; {read}



procedure writematrix(a:matrix; m,n:integer);

   var 

      i,j: integer; {counters}



   begin {write}

   for i:=1 to m do

      begin {each line}

      writeln;

      for j:=1 to n do 

         write(a[i,j]:6:2);

      end; {each line}

   writeln;

   end; {write}



procedure addmatrix(var a:matrix; b: matrix);

   var 

      i,j: integer; {counters}



   begin {addmatrix}

   for i:=1 to m do

      for j:=1 to n do 

         a[i,j]:=a[i,j]+b[i,j];

   end;{addmatrix}



procedure submatrix(var a:matrix; b: matrix);

   var 

      i,j: integer; {counters}



   begin {submatrix}

   for i:=1 to m do

      for j:=1 to n do 

         a[i,j]:=a[i,j]-b[i,j];

   end;{submatrix}



procedure menu{(var choice:char)};

   begin {menu}

   writeln;

   writeln('Do you wish to :');

   writeln('  1. Add');

   writeln('  2. Subtract');

   writeln('  3. End the program');

   readln(choice);

   end;{menu}



procedure operation(var a : matrix);

   begin {operation}

   writeln('Enter your second matrix.');

   writeln('Enter the elements of each row separated');

   writeln('by a space. Hit return at the end of each row.');

   readmatrix(b,m,n);

   if choice = 1 then

      addmatrix(a,b)

   else if choice = 2 then

      submatrix(a,b)

   else if choice = 3 then

      choice := 3

   else if ((choice <> 1) and (choice <> 2) and (choice <> 3)) then

      begin {else if}

      writeln('That was not a choice.');

      choice := 4;

      end;{else if}

   end;{operation}



begin{main program}

   clrscr;

   choice := 0;

   writeln('Enter the dimensions of the matrix ');

   writeln('separated by a space. Then hit return.');

   read(m,n);

   writeln('Enter your matrix.');

   writeln('Enter each element followed by a return.');

   writeln('Enter the first row before you go to the next row.');

   readmatrix(a,m,n);

   menu;

   while choice <> 3 do

      begin {while}

      operation(a);

      writeln('The resulting matrix is :');

      writematrix(a,m,n);

      menu

   end; {while}

   writeln('The final resulting matrix is :');

   writematrix(a,m,n);

   writeln('Press return to leave the program');

   readln;

end. {main program}