MAT101 2024 Tutorial sheet-1.pdf

Malaviya National Institute of Technology, Jaipur Department of Mathematics MAT 101 Tutorial 1- Rank of a matrix, Solution of linear simultaneous equations.  ...

Malaviya National Institute of Technology, Jaipur Department of Mathematics MAT 101 Tutorial 1- Rank of a matrix, Solution of linear simultaneous equations.   2 1 −2 1. Find the rank of the matrix A =  −1 −1 1 . 3 1 −2 2. Reduce  the following matrices   to row echelonform and hence find their ranks. 0 1 −3 −1 2 0 −1 0  1 0 1 1   4 1 0 5  (i)   3  (ii)  . 1 0 2   0 1 3 6  1 1 −2 0 6 1 −2 6 3. Reduce  the following matrices  to column echelon form and hence find their ranks. 1 1 −1 1    −1 1 −3 −3  1 −2 3 4 (i)   (ii)  −2 4 −1 −3 .  1 0 1 2  −1 2 7 6 1 −1 3 3   1 2 −1 3  4 1 2 1  4. For what value of k the matrix  3 −1 1 2  has rank 3.  1 2 0 k 5. Check if the following system of equations is consistent or inconsistent: x + y + z = 1, x + 2y + 4z = 3, x + 4y + 10z = 9. 6. Test for consistency and find the solution to the equation x + y + z = 6, x − y + 2z = 5, 3x + y + z = 8, 2x − 2y + 3z = 7. 7. For what values of λ the equations x + y + z = 1, 2x + y + 4z = λ, 4x + y + 10z = λ2 have a solution and solve them completely in each case. 8. Investigate the values of λ and µ so that the equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = µ have (i) no solution, (ii) a unique solution, and (iii) an infinite number of solutions. 9. Examine whether the following equations are consistent and solve them if they are consistent: 2x + 6y + 11 = 0, 6x + 20y − 6z + 3 = 0, 6y − 18z + 1 = 0. 10. Determine the value of λ for which the following set of equations may possess a nontrivial solution: 2x + y + 2z = 0, x + y + 3z = 0, 4x + 3y + λz = 0. Also, find the solution. 11. Determine the conditions for which the following system x + y + z = 1, x + 2y − z = b, 5x + 7y + az = b2 admits (i) unique solution, (ii) no solution, (iii) infinite solutions. 1