Mathematical Methods for Physics and Engineering - Matematica.NET

8.19 EXERCISESis 2 and that an orthogonal base for the null space of A is provided by any twocolumn matrices of the form (2 + α i − 2α i 1 α i ) T ,forwhichtheα i (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 − 3y +2z =0, 2x + y +3z =0.(b) 2x = b(y + z), x =2a(y − z), x =(6a − b)y − (6a + b)z.8.33 Solve the simultaneous equations2x +3y + z =11,x + y + z =6,5x − y +10z =34.8.34 Solve the following simultaneous equations for x 1 , x 2 and x 3 , using matrixmethods:x 1 +2x 2 +3x 3 =1,3x 1 +4x 2 +5x 3 =2,x 1 +3x 2 +4x 3 =3.8.35 Show that the following equations have solutions only if η = 1 or 2, and findthem in these cases:x + y + z =1,x +2y +4z = η,x +4y +10z = η 2 .8.36 Find the condition(s) on α such that the simultaneous equationsx 1 + αx 2 =1,x 1 − x 2 +3x 3 = −1,2x 1 − 2x 2 + αx 3 = −2have (a) exactly one solution, (b) no solutions, or (c) an infinite number ofsolutions; give all solutions where they exist.8.37 Make an LU decomposition of the matrix⎛A = ⎝ 3 6 9⎞1 0 5⎠2 −2 16and hence solve Ax = b, where(i)b = (21 9 28) T , (ii) b = (21 7 22) T .8.38 Make an LU decomposition of the matrix⎛⎞2 −3 1 3⎜1 4 −3 −3⎟A = ⎝5 3 −1 −1⎠ .3 −6 −3 1Hence solve Ax = b for (i) b =(−4 1 8 −5) T , (ii) b =(−10 0 −3 −24) T .Deduce that det A = −160 and confirm this by direct calculation.8.39 Use the Cholesky separation method to determine whether the following matricesare positive definite. For each that is, determine the corresponding lower diagonalmatrix L:⎛A =⎝ 2 1 31 3 −13 −1 1⎞ ⎛⎠ , B =313⎝ 5 0 √ ⎞3√0 3 0 ⎠ .3 0 3