Linear Algebra Question Bank PDF

## QUESTION BANK ### MODULE-1: Linear Algebra **Answer the following questions** 1. Find the rank of the following matrices by row echelon form - (i) ``` 1 2 3 4 4 3 2 -1 6 8 4 8 12 16 12 16 ``` - (ii) ``` 2 -1 -3 -1 1 0 1 1 0 1 1 - 1 4 0 2 1...

## QUESTION BANK ### MODULE-1: Linear Algebra **Answer the following questions** 1. Find the rank of the following matrices by row echelon form - (i) ``` 1 2 3 4 4 3 2 -1 6 8 4 8 12 16 12 16 ``` - (ii) ``` 2 -1 -3 -1 1 0 1 1 0 1 1 - 1 4 0 2 1 ``` - (iii) ``` 2 3 1 4 2 1 3 4 ``` - (iv) ``` A = [ 1 0 -3 -1] [ 0 1 -1 -2] [ 1 1 1 -4 ] [ 1 2 3 1 ] ``` - (v) ``` A = [ 2 3 1] [ 3 1 0] [ 1 1 1] ``` - (vi) ``` A = [ 2 3 1] [ 3 -2 1] [ 0 1 1] ``` - (vii) ``` A = [ 1 2 3 2] [ 0 2 3 4] [ 1 3 4 5] [ 2 3 5 1] ``` - (viii) ``` A = [ 2 3 5] [ 4 8 13] [ 2 3 5] [ 1 3 4] ``` 2. Test for consistency and solve the following system of equations - (i) ``` 5x + y + 3z = 20; 2x + 5y + 2z = 18; 3x + 2y + z = 14 ``` - (ii) ``` x + 2y + 3z = 14; 4x + 5y + 7z = 35; 3x + 3y + 4z = 21 ``` - (iii) ``` x+y+z=6; x-y+2z=5; 3x+y+z=8. ``` - (iv) Investigate the values of and such that the system of equations ``` x+y+z=6;x+2y+3z=10, x+2y+ z = u ``` may have - (1) Unique solution - (2) Infinite solution - (3) No solution. 3. Solve the following system of equations by Gauss elimination method - (a) ``` x - 2y + 3z = 2; 3x-y+4z = 4; 2x + y - 2z = 5 ``` - (b) ``` 3x + 4y + 5z = 18; 2x - y + z = 13; 5x - 2y + 7z = 20 ``` - (c) ``` x+y+z=9; x-2y+3z=8,2x+y-z=3. ``` - (d) ``` 2x+y=4z=12, 4x+11y-z=33, 8x-3y+2z=20. ``` 4. Solve the following system of equations by Gauss-Jordon elimination method - (a) ``` x + y + z = 8; -x-y+2z = -4; 3x + 5y - 7z = -14 ``` - (b) ``` 2x + 3y - z = 5; 4x + 4y-3z = 3; 2x - 3y+2z = 2 ``` - (c) ``` 2x+y+3z=1, 4x+4y+7z=1, 2x+5y+9z=3. ``` 5. Solve the following system of equations using Gauss Seidel method - (a) ``` 5x + 2y + z = 12; x + 4y + 2z = 15; x + 2y + 5z = 20 ``` - (b) ``` 10x + 2y + z = 9; x+10y-z-22; -2x + 3y + 10z = 22 ``` - (c) ``` 10x+y+z=12, x+10y+z=12, x+y+10z=12. ``` - (d) ``` 20x+y-2z=17, 3x+20y-z=-18,2x-3y+20z=25. ``` 6. Find the dominant Eigen value and the corresponding Eigen vector of the matrix ``` A = [ 6 -2 2 ] [ -2 3 -1 ] [ 2 -1 3 ] ``` by Rayleigh's power method taking the initial Eigen vector as [1, 1, 1]. 7. Find the numerically largest eigen value and the corresponding eigen vector of the matrix ``` A = [ 4 1 -1 ] [ 2 3 -1 ] [ 1 -2 1 ] ``` by taking the initial approximation to the eigen vector as [1, 0.8, -0.8]7. Perform 5 iterations. 8. Find the largest eigen value and the corresponding eigen vector of the matrix A by the power method given that: ``` A = [ 2 0 1 ] [ 0 2 0 ] [ 1 0 2 ] ``` 9. Find the dominant eigen value and the corresponding eigen vector of the matrix ``` A = [ 2 -1 0 ] [ -1 2 -1 ] [ 0 -1 2 ] ``` by taking the initial vector as [1,1,1]'.