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Exercise 1: Linear Algebra Tutorial 1

The document is a tutorial on linear algebra containing various exercises focused on solving systems of linear equations using Cramer’s rule, matrix operations, and determinants. It includes tasks such as finding matrix inverses, determining singular matrices, and solving linear equations through matrix representation. The exercises are structured to enhance understanding of key concepts in linear algebra.

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6 views4 pages

Exercise 1: Linear Algebra Tutorial 1

The document is a tutorial on linear algebra containing various exercises focused on solving systems of linear equations using Cramer’s rule, matrix operations, and determinants. It includes tasks such as finding matrix inverses, determining singular matrices, and solving linear equations through matrix representation. The exercises are structured to enhance understanding of key concepts in linear algebra.

Uploaded by

bourawiaboukadida422
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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Linear Algebra

Tutorial 1

Exercise 1

Exercise 2

Exercise 3

1
Exercise 4

2
Exercise 5

Exercise 6

Solve by Cramer’s rule, if it is possible, these systems of linear equations:

2𝑥 + 4𝑦 + 3𝑧 = 1
a. {3𝑥 − 6𝑦 − 2𝑧 = −2
−5𝑥 + 8𝑦 + 2𝑧 = 4

𝑥−𝑦 =0
𝒃. { −𝑥 +𝑦+𝑧 = 1
𝑥 + 2𝑦 − 3𝑧 = 0

𝑥+𝑦+𝑧 =1
𝒄. {−𝑥 + 𝑦 − 𝑧 = 2
𝑥−𝑦+𝑧 =1

Exercise 7

3
Exercise 8

Let A be the following matrix

1 3
𝐴=[ ]
−2 −8

1) Find 𝐴2 ; 𝐴−1 or show that A is not invertible.


2) Find the values of p and q such that 𝐴2 = 𝑝𝐴 + 𝑞𝐼 where 𝐼 is 2×2 identity matrix.
3) Let 𝐵 = 𝐴 − 𝑡𝐼 where t is a scalar . For which values of t is B singular.
0
4) Find the solutions to 𝐴2 𝑥 = 𝑏 where 𝑏 = [ ]
1
Exercise 9
1 2 1 7 −6 1
Let 𝐴 = (2 3 2) and 𝐵 = (−4 2 0)
3 4 5 −1 2 −1
1) Find the determinant of matrix A using two methods. Is A regular?
2) Compute 𝐴𝐵 + 2𝐼 where 𝐼 is a 3 × 3 identity matrix. Is the obtained matrix an orthogonal
matrix?
1
3) Using the previous question deduce that 𝐴−1 = − 2 𝐵
𝑥 + 2𝑦 + 𝑧 = −2
4) Let the system of linear equations : {2𝑥 + 3𝑦 + 2𝑧 = 4
3𝑥 + 4𝑦 + 5𝑧 = 8

a) Write the matrix representation for the system.


b) Solve the system using the inverse matrix method.

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