Tutorial Sheet
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI - K.K. BIRLA GOA CAMPUS Mathematics-2 Tutorial–1
1. Use the Gaussian elimination method to solve each of the following systems of linear equations: (a) −5x1 − 2x2 + 2x3 = 14 3x1 + x2 − x3 = −8 2x1 + 2x2 − x3 = −3 (b) 3x1 − 2x2 + 4x3 = −54 −x1 + x2 − 2x3 = 20 5x1 − 4x2 + 8x3 = −83 2. Find the quadratic equation y = ax2 + bx + c that goes through the points (3, 18), (2, 9) and (−2, 13). 3. Use the Gauss Jordan method to determine the complete solution set for the given system, and give one particular nontrivial solution. −2x1 − 3x2 + 2x3 − 13x4 = 0 −4x1 − 7x2 + 4x3 − 29x4 = 0 x1 + 2x2 − x3 + 8x4 = 0 4. Prove that the following homogeneous system has a nontrivial solution if and only if ad − bc = 0: ax1 + bx2 = 0 cx1 + dx2 = 0 . 5. Find all values of a for which the resulting linear system has (a) no solution, (b) a unique solution, and (c)infinitely many solutions. (i) x+y−z =2 x + 2y + z = 3 x + y + (a2 − 5)z = a (ii) x+y =3 x + (a2 − 8)y = a
6. Find an equation relating a,b and c so that the linear system: x + 2y − 3z = a 2x + 3y + 3z = b 5x + 9y − 6z = c is consistent for any values of a,b and c that satisfy that equation. 7. Show that if u and v are solution to the linear system Ax = b, then u − v is a solution to the associated homogeneous system Ax = 0. 8. If A is an n × n matrix, the homogeneous system Ax = 0 has a nontrivial solution if and only if A is singular. 9. Find all values of a for which the inverse of
1 1 0
A= 1 0 0 1 2 a exists. What is A−1 ? 10. For what values of λ does the homogeneous system (λ − 1)x + 2y = 0 2x + (λ − 1)y = 0 have a nontrivial solution?
1. Use the Gaussian elimination method to solve each of the following systems of linear equations: (a) −5x1 − 2x2 + 2x3 = 14 3x1 + x2 − x3 = −8 2x1 + 2x2 − x3 = −3 (b) 3x1 − 2x2 + 4x3 = −54 −x1 + x2 − 2x3 = 20 5x1 − 4x2 + 8x3 = −83 2. Find the quadratic equation y = ax2 + bx + c that goes through the points (3, 18), (2, 9) and (−2, 13). 3. Use the Gauss Jordan method to determine the complete solution set for the given system, and give one particular nontrivial solution. −2x1 − 3x2 + 2x3 − 13x4 = 0 −4x1 − 7x2 + 4x3 − 29x4 = 0 x1 + 2x2 − x3 + 8x4 = 0 4. Prove that the following homogeneous system has a nontrivial solution if and only if ad − bc = 0: ax1 + bx2 = 0 cx1 + dx2 = 0 . 5. Find all values of a for which the resulting linear system has (a) no solution, (b) a unique solution, and (c)infinitely many solutions. (i) x+y−z =2 x + 2y + z = 3 x + y + (a2 − 5)z = a (ii) x+y =3 x + (a2 − 8)y = a
6. Find an equation relating a,b and c so that the linear system: x + 2y − 3z = a 2x + 3y + 3z = b 5x + 9y − 6z = c is consistent for any values of a,b and c that satisfy that equation. 7. Show that if u and v are solution to the linear system Ax = b, then u − v is a solution to the associated homogeneous system Ax = 0. 8. If A is an n × n matrix, the homogeneous system Ax = 0 has a nontrivial solution if and only if A is singular. 9. Find all values of a for which the inverse of
1 1 0
A= 1 0 0 1 2 a exists. What is A−1 ? 10. For what values of λ does the homogeneous system (λ − 1)x + 2y = 0 2x + (λ − 1)y = 0 have a nontrivial solution?