8.19 EXERCISES
is 2 and that an orthogonal base for the null space ofAis provided by any two
column matrices of the form (2 +αi − 2 αi 1 αi)T,forwhichtheαi(i=1,2)
are real and satisfy 6α 1 α 2 +2(α 1 +α 2 )+5=0.
8.32 Do the following sets of equations have non-zero solutions? If so, find them.
(a) 3x+2y+z=0, x− 3 y+2z=0, 2 x+y+3z=0.
(b) 2x=b(y+z), x=2a(y−z), x=(6a−b)y−(6a+b)z.8.33 Solve the simultaneous equations
2 x+3y+z=11,
x+y+z=6,
5 x−y+10z=34.8.34 Solve the following simultaneous equations forx 1 ,x 2 andx 3 ,usingmatrix
methods:
x 1 +2x 2 +3x 3 =1,
3 x 1 +4x 2 +5x 3 =2,
x 1 +3x 2 +4x 3 =3.
8.35 Show that the following equations have solutions only ifη= 1 or 2, and find
them in these cases:
x+y+z=1,
x+2y+4z=η,
x+4y+10z=η^2.
8.36 Find the condition(s) onαsuch that the simultaneous equations
x 1 +αx 2 =1,
x 1 −x 2 +3x 3 =− 1 ,
2 x 1 − 2 x 2 +αx 3 =− 2
have (a) exactly one solution, (b) no solutions, or (c) an infinite number of
solutions; give all solutions where they exist.
8.37 Make anLUdecomposition of the matrix
A=
36 9
10 5
2 − 216
and hence solveAx=b,where(i)b= (21 9 28)T, (ii)b= (21 7 22)T.
8.38 Make anLUdecomposition of the matrix
A=
2 −31 3
14 − 3 − 3
53 − 1 − 1
3 − 6 − 31
.
Hence solveAx=bfor (i)b=(− 418 −5)T, (ii)b=(−10 0 − 3 −24)T.
Deduce that detA=−160 and confirm this by direct calculation.
8.39 Use the Cholesky separation method to determine whether the following matrices
are positive definite. For each that is, determine the corresponding lower diagonal
matrixL: