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Worksheet 3

The document contains exercises related to linear algebra, specifically focusing on solving linear systems using various methods such as matrix inversion, Cramer's rule, and Gauss elimination. It includes multiple exercises that require finding solutions for given systems of equations and proving the uniqueness of solutions. Additionally, it presents homework assignments that further explore the concepts discussed in the exercises.

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23 views1 page

Worksheet 3

The document contains exercises related to linear algebra, specifically focusing on solving linear systems using various methods such as matrix inversion, Cramer's rule, and Gauss elimination. It includes multiple exercises that require finding solutions for given systems of equations and proving the uniqueness of solutions. Additionally, it presents homework assignments that further explore the concepts discussed in the exercises.

Uploaded by

rabihhajjhassan394
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Antonine University

Computer Science Linear Algebra (Exercises) zeina.hanna@ua.edu.lb


Chapter 3: Linear Systems

2 1 1 6 𝑥
Exercise 1: Consider the matrices 𝐴 = (1 1 0), 𝐵 = (3) and 𝑋 = (𝑦).
2 1 3 8 𝑧

1. Find 𝐴−1 if it exists.


2. Using question 1, solve in ℝ the linear system 𝐴𝑋 = 𝐵.

Exercise 2: Prove that the system (𝑆) admits a unique solution and solve it in ℝ using Cramer’s rule:

−2𝑎 + 3𝑏 − 𝑐 = 1
(𝑆): { 𝑎 + 2𝑏 − 𝑐 = 4
−2𝑎 − 𝑏 + 𝑐 = −3

Exercise 3: Let 𝑎, 𝑏, 𝑐 ∈ ℝ. Solve in ℝ the following linear systems using Gauss Elimination Method:

2𝑥1 + 2𝑥2 + 3𝑥3 = 𝑎 𝑥1 + 𝑥2 + 𝑥3 = 2


(𝑆1 ): { 3𝑥1 − 𝑥2 + 5𝑥3 = 𝑏 (𝑆2 ): {2𝑥1 + 3𝑥2 + 2𝑥3 = 5
𝑥1 − 3𝑥2 + 2𝑥3 = 𝑐 2𝑥1 + 3𝑥2 + (𝑎2 − 1)𝑥3 = 𝑎 + 1

𝑥1 + 𝑥2 − 𝑥3 = 2 𝑥1 − 𝑥2 + 𝑥3 = 0
(𝑆3 ): { 1 + 2𝑥2 + 𝑥3 = 3
𝑥 (𝑆4 ): { 𝑥1 − 𝑥2 + 𝑎𝑥3 = 0
𝑥1 + 𝑥2 + (𝑎2 − 5)𝑥3 = 𝑎 𝑎𝑥1 − 2𝑎𝑥2 + (1 + 𝑎)𝑥3 = 0

Homework:
𝑥1 + 𝑥2 + 𝑥3 = 0 𝑥1 + 𝑥2 + 𝑥3 = 1
(𝑆5 ): {𝑥1 + 𝑎𝑥2 − 𝑥3 = 0 (𝑆6 ): { 2𝑥1 + 3𝑥2 + 4𝑥3 = −1
2𝑥1 + 𝑥2 + 𝑥3 = 0 𝑥1 + 2𝑥2 + (𝑎2 − 2𝑎 + 3)𝑥3 = 𝑎

1 −1 𝑥+𝑦−𝑧 2𝑥 − 𝑦 + 𝑡
Exercise 4: (Homework) Let 𝐴 = ( ). Find 𝑥, 𝑦, 𝑧 and 𝑡 such that ( ) = 𝐴.
1 0 𝑥+𝑧−𝑡 2𝑦 − 𝑧 + 2𝑡

2 1
Exercise 5: Let 𝐴 = ( ). Find all square matrices 𝑋 such that 𝑋𝐴 = 𝐴𝑋 = 𝐴.
2 1

𝑥 + (𝛼 + 1)𝑦 + 2𝑧 = 𝑎
Exercise 6: Consider the system (𝑆): {(𝛼 + 1)𝑥 + 𝑦 + 2𝑧 = 𝑏 ; 𝛼, 𝑎, 𝑏, 𝑐 ∈ ℝ.
2𝑥 + 2𝑦 + 𝛼𝑧 = 𝑐

1. For which values of 𝛼 the system doesn’t admit a unique solution?


2. Solve in ℝ the system for each of the above values of 𝛼.
3. Solve in ℝ the system when it admits a unique solution using any method.

Exercise 7: Show that ∀𝑚 ∈ ℝ, the system (𝑆) doesn’t admit a unique solution, then solve (𝑆) in ℝ.

2𝑥 + 𝑚𝑦 + 𝑧 = 3𝑚
(𝑆): {𝑥 − (2𝑚 + 1)𝑦 + 2𝑧 = 4
5𝑥 − 𝑦 + 4𝑧 = 3𝑚 − 2

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