Question

Problem ONE UseGauss-Jordan method to solve the following system of linear equations 2x - 3y +z = 0 5x + 4y +z = 10 2x - 2y - z= -1 Problem TWO Find the eigenvalues and the corresponding eigenvectors of the matrix [1 0 1 0] 0 1 1 0 0 0 20 LO 0 0 2] Problem THREE Solve the following DE x2y" - 3xy' + 4y = x2 Inx, X>0 Problem FOUR Solve the following DE y (4) - 4y" = 12 +e!

Answer 1

② from the given system of linear equations, the obtained matrix is 4 -3 1 10 on hd - 2 - -

>Re Re-Ra - 2 SR R3 R3 RI R3 R3 R2 2 5 18 을 23 5 4 3 5 تا با او O o 43 23 -43 23 Rz 23 43 R3 R2 R2 - 3 . R3. RR, - 1.R3 5 9 0 - 23 5 -23 - 2 23 Re - 5. R2 RR - 4.R2. R, .RI from this matrix, the obtained X,Y, 2 values are x = 1 Y = 1 2 = 1 Given matrix - 2 O O 2 subtract a from the diagonal entries of the matrix - - C. îor o oo 2-xo O O 2-

Find determinant of the above matrix, we get. 14 - 6x² + 131²_122+4=0 solve this equation. The roots are: 1 = 1 de=1 d3=2 X4=2 These are Eigenvalues. Next, find Eigenvectors. I d=1 Polo 2-1 O Los 2-1 Reduce this matrix >> 0 0 0 0 O ооо oo Now solve the matrix equation Vg - V3 Vu If we take v=t, q=s Vz=0, V4=0 t v= S ott S

D D=2: I-2 o I-2 > 2-2 O a o 2-1 Reduce this matrix - 1 0 до o o o o o Now, solve the matrix equation -1 oly Vq V3 If we take &z=t, V4=s. then v=t, I ₂ = t, V3=t, V₂ = 5. t V= t tt 0 - S t o S .. The Eigenvectors are t=1 eigenvectors are The Eigenvalue d=2, eigenvectors are Cili