LINEAR ALGEBRA & CALCULUS MID-I QB

1. Define the rank of the matrix and find the rank of the following matrix by using echelon form

[ ]
2 1 3 5
4 2 1 3
.
8 4 7 13
8 4 −3 −1

2. Discuss for what values of λ, μ the simultaneous equations x + y + z = 6, x +2y+ 3z = 10,

x + 2y + λ= μ have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.
3. Prove that the following equations are consistent and solve them. 3x + 3y + 2z = 1;
x + 2y = 4; 10y + 3z = -2; 2x - 3y - z = 5.

4. Solve by Gauss-elimination method. 2x1 + x2 + 2x3 + x4 = 6; 6x1 - 6x2 + 6x3 + 12x4 = 36;
4x1 + 3x2 + 3x3 - 3x4 =-1; 2x1 + 2x2 - x3 + x4 = 10.
5. Use the Gauss – Seidel iteration method to solve the system. 10x + y + z = 12;
2x + 10y + z = 13; 2x + 2y + 10z = 14.

6. Solve the system of equations x + 3y - 2z = 0; 2x – y + 4z = 0; x - 11y + 14z = 0.

7. Using Jacobi’s iteration method to solve the following equations 10x + 2y + z = 9; x + 10y –

z = -22; -2x + 3y + 10z = 22.

[ ]
1 2 3 0
2 4 3 2
8. Reduce the matrix A= into echelon form and hence find its rank.
3 2 1 3
6 8 7 5

[ ]
2 −2 0 6
4 2 0 2
9. find the rank of the following matrix A= by reducing it to
1 −1 0 3
1 −2 1 2
canonical form.
10. Solve the system of equations by using Gauss-Seidel iterative method
x + 10y + z = 6;10x + y + z = 6; x + y + 10z = 6.

[ ]
1 2 3 0
2 4 3 2
11. Reduce A to canonical form and find its rank, if A=
3 2 1 3
6 8 7 5

12. Solve the equations 5x – y + 3z = 10; 3x + 6y =18; x + y + 5z = -10 by Jacobi’s

method with (3,0, -2) as the initial approximation to the solution.

[ ]
8 −6 2
13. Find the eigen values and the corresponding eigen vectors −6 7 −4 .
2 −4 3
 [ ]
3 −1 1
14. Diagonalize the matrix A = −1 5 −1 and find A4 .
1 −1 3

[ ]
8 −8 2
15. Verify Cayley-Hamilton theorem for the matrix A = 4 −3 −2 and find A-1 & A8.
3 −4 1

[ ]
8 −6 2
16. Find the eigen values and the corresponding eigen vectors −6 7 −4 .
2 −4 3

[ ]
1 1 3
17. Verify Cayley-Hamilton theorem for the matrix A = 1 3 −3 and find A-1 .
−2 −4 −4

[ ]
8 −8 −2
18. Diagonalize the matrix A= 4 −3 −2 .
3 −4 1

[ ]
6 −2 2
19. Find the E.Values of the matrix −2 3 −1 and the corresponding eigen vectors.
2 −1 3

20. Determine the eigen values and eigen vectors of B=2A2-1/2A+3I Where A = [ 82 −42 ]
[ ]
1 0 3
21. Verify Cayley-Hamilton theorem and find the inverse of 2 −1 −1 .
1 −1 1

22. Using Cayley-Hamilton theorem, find A8, if A = [ 12 −12 ] .

[ ]
−9 4 4
23. Show that the matrix A= −8 3 4 is diagonalizable
−16 8 7

[ ]
6 −2 2
24. Find the characteristic roots of the matrix −2 3 −1 and the corresponding eigen
2 −1 3
vectors.
25. Reduce the quadratic form to the canonical form 3x2+2y2+3z2-2xy-2yz.
26. Reduce the quadratic form 3x2+3y2+3z2+4xy+8yz+8xz into canonical form by
diagonalization. Find its nature, rank, index and signature.
27. Reduce the quadratic form 8x2+7y2+3z2-12xy-8yz+4xz into canonical form.

[ ]
6 −2 2
28. Find the eigen vectors of the matrix −2 3 −1 and hence reduce 6x2+3y2+3z2-
2 −1 3
2yz+4zx-4xy to a sum of squares.
29. Reduce the quadratic form to canonical form by an orthogonal reduction
 And state the nature of the quadratic form 5x2+26y2+10z2+4yz+14zx+6xy.

30. Reduce the quadratic form to the canonical form 2x2+2y2+2z2-2xy+2zx-2yz.