1. The document discusses matrices, which are rectangular arrays of elements arranged in rows and columns. Common matrix operations include addition and multiplication.
2. Matrices come in several forms, including square, rectangular, null, diagonal, scalar, and identity matrices. Special types of square matrices include triangular superior and triangular inferior matrices.
3. For matrices to be equal, the elements in each corresponding position must be equal. Matrix operations like addition involve performing the operation on corresponding elements.
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
* Plot ordered pairs in a Cartesian coordinate system.
* Graph equations by plotting points.
* Find x-intercepts and y-intercepts.
* Use the distance formula.
* Use the midpoint formula.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
3. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝑹𝒆𝒄𝒐𝒏𝒐𝒄𝒆𝒓 𝒍𝒐𝒔
𝒆𝒍𝒆𝒎𝒆𝒏𝒕𝒐𝒔 𝒅𝒆
𝒖𝒏𝒂 𝒎𝒂𝒕𝒓𝒊𝒛
𝑹𝒆𝒂𝒍𝒊𝒛𝒂𝒓 𝒍𝒂𝒔
𝒐𝒑𝒆𝒓𝒂𝒄𝒊𝒐𝒏𝒆𝒔
𝒄𝒐𝒏 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔
𝑼𝒕𝒊𝒍𝒊𝒛𝒂𝒓 𝒍𝒂𝒔
𝒑𝒓𝒐𝒑𝒊𝒆𝒅𝒂𝒅𝒆𝒔
𝒅𝒆 𝒍𝒂𝒔 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔.
4. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
MATRICES
En 1848 James Joseph Sylvester introduce el
término “matriz”, que significara la madre de
las determinantes.
El uso de las matrices es más antiguo, esto se
observa en la resolución de sistemas de
ecuaciones lineales.
Sus aplicaciones se realizan para cuando se
tiene una gran cantidad de elementos que se
tiene a trabajar, como en programas de base
de datos SQL, Excel; al igual que en los
programas de programación.
Toda imagen digital es una matriz y a partir
de la evolución de la programación, se tiene
entre sus aplicaciones los videojuegos, que
han ido evolucionando desde los juegos 2D
hasta los 3D, mejorando la imagen gráfica y la
dinámica del mismo.
Primera versión del juego Final Fantasy del año 1987
5. C R E E M O S E N L A E X I G E N C I A
MATRIZ
C U R S O D E Á L G E B R A
Una matriz es un arreglo rectangular de elementos
distribuidos en filas y columnas.
Ejemplos:
𝐹𝑖𝑙𝑎
𝐶𝑜𝑙𝑢𝑚𝑛𝑎
𝑀 =
1 6 0
𝜋 5 7
1
−9
4 −2 5 −7
La matriz M tiene 3 filas y 4 columnas
𝑁 =
−3
2
5
0
9
7
−1
4
La matriz N tiene 4 filas y 2 columnas
En general:
𝐴 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
⋯
𝑎1𝑛
𝑎2𝑛
𝑎3𝑛
⋮ ⋱ ⋮
𝑎𝑚1 𝑎𝑚2 𝑎𝑚3 ⋯ 𝑎𝑚𝑛
𝐹1
𝐹2
𝐹3
⋮
𝐹𝑚
𝑁°𝑑𝑒 𝑓𝑖𝑙𝑎
𝑁°𝑑𝑒 𝑐𝑜𝑙𝑢𝑚𝑛𝑎
C1 C2 C3 Cn
Notación: 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
𝑛ú𝑚𝑒𝑟𝑜 𝑑𝑒 𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑠
𝑛ú𝑚𝑒𝑟𝑜 𝑑𝑒 𝑓𝑖𝑙𝑎𝑠
𝑂𝑟𝑑𝑒𝑛
𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑙𝑎
𝑓𝑖𝑙𝑎 𝑖, 𝑐𝑜𝑙𝑢𝑚𝑛𝑎 𝑗
6. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
ELEMENTO DE UNA MATRIZ (𝒂𝒊𝒋)
Los elementos se ubican por fila y columna.
El elemento 𝑎𝑖𝑗 se ubica en la 𝑓𝑖𝑙𝑎 𝑖, 𝑐𝑜𝑙𝑢𝑚𝑛𝑎 𝑗
Ejemplo:
Dada la matriz A = 𝑎𝑖𝑗 𝑚×𝑛
𝐴 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
Luego:
𝑎11 = −3 𝑎12 = 4 𝑎13 = 𝜋
𝑎21 = 7 𝑎22 = −8 𝑎23 = 5
𝑎31 = 0 𝑎32 = 1 𝑎33 = 2
Aplicación:
Encuentre la matriz A = 𝑎𝑖𝑗 3×3
donde
𝑎𝑖𝑗 = ൝
𝑖𝑗 ; 𝑖 ≥ 𝑗
𝑖 − 𝑗 ; 𝑖 < 𝑗
Como A = 𝑎𝑖𝑗 3×3
, se tiene:
𝑖 ≥ 𝑗
𝑖 < 𝑗
→ 𝐴 =
−3 4 𝜋
7 −8 5
0 1 2
𝐴 =
1 −1 −2
2 3 −1
3 6 9
7. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
MATRICES ESPECIALES
𝟏) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓:
𝑁° 𝑑𝑒 𝑓𝑖𝑙𝑎𝑠 ≠ 𝑁° 𝑑𝑒 𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑠
𝑀 =
1 6 0
𝜋 5 7
1
−9
4 −2 5 −7
𝟐) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒄𝒖𝒂𝒅𝒓𝒂𝒅𝒂:
𝑁° 𝑑𝑒 𝑓𝑖𝑙𝑎𝑠 = 𝑁° 𝑑𝑒 𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑠
• M tiene 3 filas y 4 columnas
• El orden de la matriz M es 3 × 4
• 𝑀 = 𝑚𝑖𝑗 3×4
• 𝑀 ∈ ℝ3×4
Ejemplo:
Ejemplo:
−3 4 𝜋
7 −8 5
0 1 2
𝐴 =
• A tiene 3 filas y 3 columnas
• El orden de la matriz cuadrada A es 3
• 𝐴 = 𝑎𝑖𝑗 3×3
• 𝐴 ∈ ℝ3×3
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑠𝑒𝑐𝑢𝑛𝑑𝑎𝑟𝑖𝑎
8. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝟑) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒏𝒖𝒍𝒂:
𝑎𝑖𝑗 = 0 ∀ 𝑖, 𝑗
Ejemplos:
𝐴 =
0 0 0
0 0 0
0
0
0 0 0 0
0 0 0
0 0 0
0 0 0
𝐵 =
Nota:
• 𝐿𝑎 𝑚𝑎𝑡𝑟𝑖𝑧 𝑛𝑢𝑙𝑎 𝑝𝑢𝑒𝑑𝑒 𝑠𝑒𝑟
𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑜 𝑐𝑢𝑎𝑑𝑟𝑎𝑑𝑎
𝟑) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒅𝒊𝒂𝒈𝒐𝒏𝒂𝒍:
Es una matriz cuadrada, donde se cumple:
𝑎𝑖𝑗 = 0; ∀ 𝑖 ≠ 𝑗
Ejemplos:
3 0 0
0 0 0
0 0 𝜋
𝐴 =
0 0 0
0 0 0
0 0 0
𝐵 =
𝐴 = 𝑑𝑖𝑎𝑔(3; 0; 𝜋) 𝐵 = 𝑑𝑖𝑎𝑔(0; 0; 0)
Nota:
• 𝐿𝑎 𝑚𝑎𝑡𝑟𝑖𝑧 𝑛𝑢𝑙𝑎 𝑡𝑎𝑚𝑏𝑖é𝑛 𝑒𝑠
𝑢𝑛𝑎 𝑚𝑎𝑡𝑟𝑖𝑧 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙
9. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝟑) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒆𝒔𝒄𝒂𝒍𝒂𝒓:
Es una matriz cuadrada, donde se cumple:
𝑎𝑖𝑗 = ൝
0 ; ∀ 𝑖 ≠ 𝑗
𝑘 ; 𝑖 = 𝑗
; 𝑘 ∈ ℝ
Ejemplos:
5 0 0
0 5 0
0 0 5
𝑀 =
𝑁 =
−1 0
0 −1
0 0
0 0
0 0
0 0
−1 0
0 −1
𝟒) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒊𝒅𝒆𝒏𝒕𝒊𝒅𝒂𝒅 (𝑰):
Es una matriz cuadrada, donde se cumple:
𝑎𝑖𝑗 = ൝
0 ; ∀ 𝑖 ≠ 𝑗
1 ; 𝑖 = 𝑗
Ejemplos:
1 0 0
0 1 0
0 0 1
𝐹 =
𝐺 =
1 0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
= 𝐼3
= 𝐼4
10. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝟓) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒕𝒓𝒊𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒔𝒖𝒑𝒆𝒓𝒊𝒐𝒓:
Es una matriz cuadrada, donde se cumple:
𝑎𝑖𝑗 = 0 ; ∀ 𝑖 > 𝑗
Ejemplos:
0 2 4
0 7 5
0 0 3
𝐴 =
𝐵 =
4 6
0 7
8 9
0 1
0 0
0 0
6 0
0 3
𝟔) 𝑴𝒂𝒕𝒓𝒊𝒛 𝒕𝒓𝒊𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒊𝒏𝒇𝒆𝒓𝒊𝒐𝒓:
Es una matriz cuadrada, donde se cumple:
𝑎𝑖𝑗 = 0 ; ∀ 𝑖 < 𝑗
Ejemplos:
0 0 0
2 7 0
0 5 3
𝐶 =
𝐷 =
4 0
0 7
0 0
0 0
2 0
0 7
6 0
1 3
11. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
IGUALDAD DE MATRICES
Sean las matrices A = 𝑎𝑖𝑗 𝑚×𝑛
𝑦 B = 𝑏𝑖𝑗 𝑚×𝑛
entonces
𝐴 = 𝐵 ⟺ 𝑎𝑖𝑗 =𝑏𝑖𝑗 ; ∀ 𝑖, 𝑗
Ejemplo:
Sean las matrices
𝐴 = 5 𝑚 4
𝑛 8 6
𝑦 𝐵 = 5 7 4
3 8 6
son iguales, entonces:
𝑚 = 7 ∧ 𝑛 = 3
OPERACIONES CON MATRICES
Adición:
Sean las matrices A = 𝑎𝑖𝑗 𝑚×𝑛
𝑦 B = 𝑏𝑖𝑗 𝑚×𝑛
𝐴 + 𝐵 = (𝑎𝑖𝑗+𝑏𝑖𝑗) ; ∀ 𝑖, 𝑗
Ejemplo:
Sean las matrices
𝐴 = 1 4 0
0 8 7
𝑦 𝐵 = 2 3 5
3 1 2
entonces
𝐴 + 𝐵 =
3 7 5
3 9 9
𝑚 × 𝑛
12. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Multiplicación de matrices:
• Multiplicación de un escalar por una matriz:
Sea la matriz A = 𝑎𝑖𝑗 𝑚×𝑛
y 𝑘 un número real
𝑘. 𝐴 = 𝑘. 𝑎𝑖𝑗 𝑚×𝑛
Ejemplo:
Dada la matriz 𝐴 =
1 0 −1
2 1 3
entonces:
4𝐴 =
4 0 −4
8 4 12
• Multiplicación de una matriz fila por una matriz
columna
Sean las matrices 𝐴 = 𝑎1 𝑎2 … 𝑎𝑛 y 𝐵 =
𝑏1
𝑏2
⋮
𝑏𝑛
Se define su multiplicación como
𝐴. 𝐵 = 𝑎1𝑏1 + 𝑎2𝑏2 + ⋯ + 𝑎2𝑏2
Ejemplo:
Si 𝐴 = 2 −1 −2 y 𝐵 =
−3
4
−6
entonces:
𝐴. 𝐵 = (2)(−3) + (−1)(4)+ (−2)(−6)
𝐴. 𝐵 = 2
13. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
• Multiplicación de dos matrices:
Sean las matrices A = 𝑎𝑖𝑗 𝑚×𝑛
y B = 𝑏𝑖𝑗 𝑛×𝑝
se define:
𝐴𝐵 = 𝑐𝑖𝑘 𝑚×𝑝 tal que 𝑐𝑖𝑘 =
𝑗=1
𝑛
𝑎𝑖𝑗𝑏𝑗𝑘
Ejemplo:
Si 𝐴 =
1 −1
2 0
y 𝐵 =
4 0 −2
1 3 −1
entonces
𝐴𝐵 =
1 −1
2 0
4 0 −2
1 3 −1
𝐴𝐵 =
1 −1
4
1
1 −1
0
3
1 −1
−2
−1
2 0
4
1
2 0
0
3
2 0
−2
−1
Luego
𝐴𝐵 =
3 −3 −1
8 0 −4
Nota:
𝑃𝑎𝑟𝑎 𝑝𝑜𝑑𝑒𝑟 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑟 𝐴 𝑝𝑜𝑟 𝐵, 𝑒𝑙 𝑛ú𝑚𝑒𝑟𝑜 𝑑𝑒
𝑐𝑜𝑙𝑢𝑚𝑛𝑎𝑠 𝑑𝑒 𝐴 𝑑𝑒𝑏𝑒 𝑠𝑒𝑟 𝑖𝑔𝑢𝑎𝑙 𝑎 𝑙𝑎 𝑐𝑎𝑛𝑡𝑖𝑑𝑎𝑑 𝑑𝑒
𝑓𝑖𝑙𝑎𝑠 𝑑𝑒 𝐵
Si A = 𝑎𝑖𝑗 3×4
; B = 𝑏𝑖𝑗 4×5
→ 𝐴𝐵 = 𝑐𝑖𝑗 3×5
14. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Ejemplo:
Si 𝐴 =
1 0 −1
2 1 −2
0 −1 −1
y 𝐵 =
2 1 −2
1 0 −1
1 1 −2
Calcule A. B y B. A
𝐴 =
1 0 −1
2 1 −2
0 −1 −1
𝐵 =
2 1 −2
1 0 −1
1 1 −2
1 0 0
3 0 −1
−2 −1 3
= 𝐴𝐵
Entonces:
𝐴𝐵 =
1 0 0
3 0 −1
−2 −1 3
Además:
𝐵 =
2 1 −2
1 0 −1
1 1 −2
𝐴 =
1 0 −1
2 1 −2
0 −1 −1
4 3 −2
1 1 0
3 3 −1
= 𝐵𝐴
Entonces:
𝐵𝐴 =
4 3 −2
1 1 0
3 3 −1
Se nota que: 𝐴𝐵 ≠ 𝐵𝐴
15. C R E E M O S E N L A E X I G E N C I A
Propiedades:
C U R S O D E Á L G E B R A
𝐴 + 𝐵
𝑆𝑖𝑒𝑛𝑑𝑜 𝐴 𝑦 𝐵 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠, 𝑡𝑒𝑛𝑒𝑚𝑜𝑠
1) = 𝐵 + 𝐴
2) 𝐴 + 𝐵 + 𝐶 = 𝐴 + (𝐵 + 𝐶)
3) 𝐸𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝐴𝐵 ≠ 𝐵𝐴
• 𝑆𝑖 𝐴𝐵 = 𝐵𝐴 𝑠𝑒 𝑑𝑖𝑐𝑒 𝑞𝑢𝑒 𝐴 𝑦 𝐵
𝑠𝑜𝑛 𝑐𝑜𝑛𝑚𝑢𝑡𝑎𝑏𝑙𝑒𝑠.
• 𝑆𝑒 𝑐𝑢𝑚𝑝𝑙𝑒 𝑞𝑢𝑒 𝐴𝐼 = 𝐼𝐴 = A
𝑑𝑜𝑛𝑑𝑒 𝐼 𝑒𝑠 𝑙𝑎 𝑚𝑎𝑡𝑟𝑖𝑧 𝑖𝑑𝑒𝑛𝑡𝑖𝑑𝑎𝑑.
• 𝑆𝑒 𝑐𝑢𝑚𝑝𝑙𝑒 𝑞𝑢𝑒 𝐴Θ = Θ𝐴 = Θ
𝑑𝑜𝑛𝑑𝑒 Θ 𝑒𝑠 𝑙𝑎 𝑚𝑎𝑡𝑟𝑖𝑧 𝑛𝑢𝑙𝑎.
4) 𝐴. 𝐵 . 𝐶 = 𝐴. (𝐵. 𝐶)
5) 𝐴 + 𝐵 . 𝐶 = 𝐴𝐶 + 𝐵𝐶
6) 𝐴 𝐵 + 𝐶 = 𝐴𝐵 + 𝐴𝐶
Ejemplos:
𝐼𝑛𝑑𝑖𝑞𝑢𝑒 𝑣𝑒𝑟𝑑𝑎𝑑𝑒𝑟𝑜 𝑉 𝑜 𝑓𝑎𝑙𝑠𝑜(𝐹)
𝑒𝑛 𝑐𝑎𝑑𝑎 𝑐𝑎𝑠𝑜.
1) 𝑆𝑖 𝐴𝐵 = Θ → 𝐴 = Θ ∨ 𝐵 = Θ
𝐸𝑠 𝑓𝑎𝑙𝑠𝑜, 𝑝𝑢𝑒𝑠𝑡𝑜 𝑞𝑢𝑒
1 0
0 0
0 0
0 1
=
0 0
0 0
൝
𝐴
൝
𝐵
൝
Θ
2) 𝑆𝑖 𝐴𝐵 = 𝐴𝐶 → 𝐵 = 𝐶
𝐸𝑠 𝑓𝑎𝑙𝑠𝑜, 𝑝𝑢𝑒𝑠𝑡𝑜 𝑞𝑢𝑒
Θ𝐵 = Θ𝐶 = Θ
3) 𝑆𝑖𝑒𝑛𝑑𝑜 𝐴 𝑦 𝐵 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠
𝑐𝑢𝑎𝑑𝑟𝑎𝑑𝑎𝑠 𝑑𝑒𝑙 𝑚𝑖𝑠𝑚𝑜 𝑜𝑟𝑑𝑒𝑛
(𝐴 + 𝐵)2= 𝐴2 + 2𝐴𝐵 + 𝐵2
𝐸𝑠 𝑓𝑎𝑙𝑠𝑜, 𝑝𝑢𝑒𝑠𝑡𝑜 𝑞𝑢𝑒
(𝐴 + 𝐵)2
= (𝐴 + 𝐵)(𝐴 + 𝐵)
= 𝐴𝐴 +𝐴𝐵 +𝐵𝐴 +𝐵𝐵
= 𝐴2 +𝐴𝐵 +𝐵𝐴 +𝐵2
Nota:
• 𝑆𝑖 𝐴 𝑦 𝐵 𝑠𝑜𝑛 𝑐𝑜𝑛𝑚𝑢𝑡𝑎𝑏𝑙𝑒𝑠
→(𝐴 + 𝐵)2= 𝐴2 + 2𝐴𝐵 + 𝐵2
16. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
• Potenciación de matrices:
Si A es una matriz cuadrada, se define:
𝐴𝑛 = 𝐴. 𝐴. 𝐴 … 𝐴
൞
𝑛 𝑣𝑒𝑐𝑒𝑠
; 𝑛 ∈ ℕ
Además:
𝐴0 = 𝐼 𝐴1 = 𝐴
• Donde I es la matriz identidad
Ejemplo:
Si 𝐴 =
1 −1
−1 0
calcule 𝐴4
Luego:
𝐴 =
1 −1
−1 0
𝐴 =
1 −1
−1 0
2 −1
−1 1
𝐴2
1 −1
−1 0
3 −2
−2 1
𝐴3
1 −1
−1 0
5 −3
−3 2
𝐴4
𝐴4
=
5 −3
−3 2
Matrices especiales
• Matriz idempotente 𝐴2 = 𝐴
• Matriz involutiva 𝐴2 = 𝐼
• Matriz nilpotente 𝐴𝑚 = Θ
, I es la matriz identidad
, Θ es la matriz nula
donde 𝑚 ∈ ℤ+
, 𝑚 es el grado de nilpotencia.
17. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
TRANSPUESTA DE UNA MATRIZ
Dada la matriz A = 𝑎𝑖𝑗 𝑚×𝑛
su transpuesta es 𝐴𝑇
definida por: 𝐴𝑇
= 𝑎𝑗𝑖 𝑛×𝑚
Es decir, se intercambiaron las filas por las columnas
Ejemplos:
𝐴 = → 𝐴𝑇 =
1 3
−2 5
0 6
1 −2 0
3 5 6
2× 3 3× 2
𝐵 =
1 2 3
4 5 6
7 8 9 3× 3
→ 𝐵𝑇
=
1 4 7
2 5 8
3 6 9 3× 3
Propiedades:
1) 𝐴𝑇 𝑇 = 𝐴
2) 𝑘. 𝐴 𝑇 = 𝑘. 𝐴𝑇 ; 𝑘 ∈ ℝ
3) 𝐴 + 𝐵 𝑇 = 𝐴𝑇 + 𝐵𝑇
4) 𝐴. 𝐵 𝑇
= 𝐵𝑇
. 𝐴𝑇
5) 𝐴𝑛 𝑇 = 𝐴𝑇 𝑛 ; 𝑛 ∈ ℕ
6) 𝐴−1 𝑇
= 𝐴𝑇 −1
Matrices especiales
• Matriz simétrica
𝐴𝑇
= 𝐴
Ejemplo:
−3 1 8
1 −7 5
8 5 2
𝐴 =
• Matriz antisimétrica
𝐴𝑇
= −𝐴
Ejemplo:
0 −1 8
1 0 5
−8 −5 0
𝐴 =
18. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
TRAZA DE UNA MATRIZ
Dada una matriz cuadrada A, su traza denotada por
Traz(A) es la suma de los elementos de su diagonal
principal.
Ejemplo:
−3 2 𝜋
7 −1 6
0 3/2 5
𝐴 =
Traz(A) = −3 − 1 + 5 = 1
𝐵 =
4 −3
2 −1
Traz(B) = 4 − 1 = 3
Propiedades:
Sean A y B matrices cuadradas del mismo orden, luego
1) Traz(𝐴𝑇) = Traz(A)
Traz(k. A) = k. Traz(A) ; 𝑘 ∈ ℝ
2)
Traz(A + B) = Traz(A) + Traz(B)
3)
4) Traz(A. B) = Traz(B. A)
Ejemplo:
Se tiene 𝐴𝑇
= 2𝐴 +
3 4
5 6
. Calcule Traz(A)
Tomando Traza en ambos lados
𝐴𝑇
= 2𝐴 +
3 4
5 6
𝑇𝑟𝑎𝑧 𝑇𝑟𝑎𝑧 𝑇𝑟𝑎𝑧
൝
𝑇𝑟𝑎𝑧(𝐴)
൝
2𝑇𝑟𝑎𝑧(𝐴) 9
= + → 𝑇𝑟𝑎𝑧(𝐴) = −9