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Tutorial 2

This document outlines the tutorial for the Linear Algebra & ODE course at Bennett University for the academic year 2024-25. It includes a series of questions related to reduced row echelon forms, solving systems of linear equations using various methods, and conditions for the existence of solutions. The tutorial emphasizes practical applications of linear algebra concepts through problem-solving.

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40 views2 pages

Tutorial 2

This document outlines the tutorial for the Linear Algebra & ODE course at Bennett University for the academic year 2024-25. It includes a series of questions related to reduced row echelon forms, solving systems of linear equations using various methods, and conditions for the existence of solutions. The tutorial emphasizes practical applications of linear algebra concepts through problem-solving.

Uploaded by

kinghero11211
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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School of Computer Science Engineering and Technology

Bennett University
Course Name: Linear Algebra & ODE Course Code: EMAT102L
Academic Year: 2024-25 Semester: Even
Date: 13/01/2025 Type: Core (L-T-P: 3-1-0)
CO1 CO2 CO3 CO4 CO5 CO6
Q1 ✓
Q2 ✓
Q3 ✓
Q4 ✓
Q5 ✓
Q6 ✓

Tutorial-2
1. What are the possible reduced row echelon forms of each of a 2 × 2 and a 3 × 3 matrix?

2. Find the row echelon form of each of the following matrices. Further, reduce them into
reduced row echelon form:
     
1 −1 2 3 1 2 3 4 3 5 −6
4
     
0 5 6 2 5 6 8 7 2 3 1 1 
,  , and .
     
  
−1 2 4 3 9 10 11 12 0 2 0 0 
     
1 2 −1 2 13 14 15 16 5 −5 5 5

3. Solve the following two systems of linear equations, both of them have the same matrix
of coefficients:

x1 − x2 + 3x3 = b1 , 2x1 − x2 + 4x3 = b2 , −x1 + 2x2 − 4x3 = b3 ,

for [b1 , b2 , b3 ]t = [0, 1, 2]t , and [3, 3, −4]t .

4. Solve the following systems of equations using Gaussian elimination method as well as
Gauss-Jordan elimination method, whenever they are consistent:
(a) x + y + z = 3, x − y − z = −1, 4x + 4y + z = 9;

(b) −x + y + z + w = 0, x − y + z + w = 0, −x + y + 3z + 3w = 0, x − y + 5z + 5w = 0;

(c) x + y + 2z = 3, −x − 3y + 4z = 2, −x − 5y + 10z = 11;

(d) 2w + 3x − y + 4z = 0, 3w − x + z = 1, 3w − 4x + y − z = 2.

5. For what values of λ ∈ R and k ∈ R, the following systems of equations have:

(i.) no solution,

(ii.) a unique solution,

(iii.) infinitely many solutions?

(a) x + y + z = 3, x + 2y + λz = 4, 2x + 3y + 2λz = k;

(b) x + y + 2z = 3, x + 2y + λz = 5, x + 2y + 4z = k.

Also, find the solutions whenever they exist.

6. Let A be an n × n matrix. If the system A2 x = 0 has a non-trivial solution, then show


that the system Ax = 0 also has a non-trivial solution.

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