# 1) Solve the matrix for a linear system.
# 
# 
> with(linalg):
Warning: new definition for   norm
Warning: new definition for   trace

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> S1:= linalg[matrix] (3,6,[0, 4,-1,3,7,1, 0,0,0,3,4,9, 0,0,0,0,5,0] );

                                 [ 0  4  -1  3  7  1 ]
                                 [                   ]
                           S1 := [ 0  0   0  3  4  9 ]
                                 [                   ]
                                 [ 0  0   0  0  5  0 ]
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> eqns := {4*y-z+3*w+7*v=1,  3*w+4*v=9,  5*v=0};

            eqns := {4 y - z + 3 w + 7 v = 1, 3 w + 4 v = 9, 5 v = 0}
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> S1 := solve (eqns);

                    S1 := {v = 0, z = 4 y + 8, w = 3, y = y}
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# 
# 2) Solve the system by Gaussian Elimination.
# 
# 
> eqns := { 3*x1-x2+x3-5*x4-x5=0,  6*x1-2*x2+2*x3-9*x4+x5=0,  (-9)*x1+3*x2-3*x3+11*x4-x5=0 };

  eqns := {3 x1 - x2 + x3 - 5 x4 - x5 = 0, 6 x1 - 2 x2 + 2 x3 - 9 x4 + x5 = 0,

      - 9 x1 + 3 x2 - 3 x3 + 11 x4 - x5 = 0}
> printlevel := <3>
> S2 := solve(eqns);

            S2 := {x5 = 0, x4 = 0, x2 = 3 x1 + x3, x1 = x1, x3 = x3}
> printlevel := <3>\

# !!!   Here is problem 24 !!!
# 
> eqns := { x1+x2+x3+x4+x5=5, x1+x5=-4, x1-x2=3};

         eqns := {x1 + x2 + x3 + x4 + x5 = 5, x1 + x5 = -4, x1 - x2 = 3}
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> S3 := solve(eqns);

    S3 := {x1 = - x5 - 4, x2 = - x5 - 7, x3 = 16 + x5 - x4, x4 = x4, x5 = x5}
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# 
# 3)  Explain:
# 
# a).  if a = 0, then no multiple of equation (1) would be subtracted from equation (3).
#           This is true because when a = 0, there is no need of elimination.
# 
# b). If b = 0, then no multiple of equation (2) would be subtracted from equation (3).
#           If a!=0, we need to subtract a multiple of equation (1) which is a scalor s*eqn(1) from eqn(3) to eliminate a, after the 
# first column elimination, b becomes b-s*3 != 0, so this is false.
#          Else if a=0, this is true.
# 
# c). If a = 0 and b = 0, then no multiple of equation (1) or equation (2) would be subtracted from equation (3).
#           Same reason with a) and b), this is true.  Because no non-zero coefficient, no elimination.