I.

INTRODUCTION

MATRIX AND ITS SIGNIFICANCE

A MATRIX IS A BASIC MATHEMATICAL CONCEPT THAT REPRESENTS A SET OF NUMBERS
ARRANGED IN ROWS AND COLUMNS. IT IS WIDELY USED IN DIFFERENT AREAS SUCH AS
MATHEMATICS, PHYSICS, ENGINEERING, AND COMPUTER SCIENCE. IN LINEAR ALGEBRA,
MATRICES ARE PARTICULARLY IMPORTANT FOR SOLVING LINEAR EQUATIONS,
MANIPULATING GEOMETRIC SHAPES, AND STUDYING COMPLEX DATA.

UNDERSTANDING MATLAB
MATLAB, WHICH STANDS FOR MATRIX LABORATORY, IS A PROGRAMMING LANGUAGE
AND INTERACTIVE ENVIRONMENT THAT FOCUSES ON NUMERICAL COMPUTATIONS AND
DATA VISUALIZATION. IT IS WIDELY USED IN ACADEMIC AND INDUSTRIAL SETTINGS FOR A
VARIETY OF TASKS, INCLUDING ALGORITHM DEVELOPMENT, SIMULATION, AND MODELING.
MATLAB'S USER-FRIENDLY INTERFACE AND WIDE RANGE OF CAPABILITIES MAKE IT AN
ESSENTIAL TOOL FOR ENGINEERS, SCIENTISTS, AND RESEARCHERS IN VARIOUS FIELDS.

SIGNIFICANCE OF MATRIX OPERATIONS

PERFORMING OPERATIONS ON MATRICES IS ESSENTIAL IN VARIOUS MATHEMATICAL
APPLICATIONS. DETERMINANTS, RANKS, TRACES, AND INVERSES PROVIDE VALUABLE
INSIGHTS INTO THE PROPERTIES AND BEHAVIOR OF MATRICES. EIGENVALUES AND
EIGENVECTORS, ON THE OTHER HAND, REVEAL IMPORTANT INFORMATION ABOUT LINEAR
TRANSFORMATIONS AND STABILITY IN DYNAMIC SYSTEMS. UNDERSTANDING THESE
OPERATIONS IS CRUCIAL FOR ADVANCED STUDIES IN ENGINEERING AND MATHEMATICS.

IN THIS REPORT, WE WILL ANALYZE TWO MATRICES, A AND B, USING MATLAB TO

ADDRESS SPECIFIC QUESTIONS AND PERFORM OPERATIONS. CHAPTER V WILL FOCUS ON
ANSWERING THE PROVIDED QUESTIONS RELATED TO MATRIX B.

II. REVIEW RELATED LITERATURE

MATRIX THEORY AND APPLICATIONS

MATRIX THEORY, A FUNDAMENTAL ASPECT OF LINEAR ALGEBRA, HAS EXTENSIVE
APPLICATIONS IN VARIOUS FIELDS. WHETHER IN ENGINEERING, PHYSICS, ECONOMICS, OR
COMPUTER SCIENCE, MATRICES PLAY A VITAL ROLE IN SOLVING INTRICATE PROBLEMS. THE
WORKS OF STRANG (2005) AND LUENBERGER (2008) OFFER COMPREHENSIVE INSIGHTS
INTO THE THEORETICAL FOUNDATIONS AND PRACTICAL USES OF MATRICES.

MATLAB AS A COMPUTATIONAL TOOL

MATLAB IS WIDELY RECOGNIZED AS A LEADING COMPUTATIONAL TOOL FOR
ENGINEERS AND SCIENTISTS. WITH ITS EXTENSIVE RANGE OF FUNCTIONS AND USER-
FRIENDLY SYNTAX, IT ENABLES NUMERICAL ANALYSIS, SIMULATION, AND ALGORITHM
DEVELOPMENT. GILAT'S WORK (2014) SERVES AS A VALUABLE RESOURCE FOR BEGINNERS,
PROVIDING A COMPREHENSIVE INTRODUCTION TO MATLAB'S CAPABILITIES AND PRACTICAL
APPLICATIONS.

MATRIX OPERATIONS AND PROPERTIES

 UNDERSTANDING THE BEHAVIOR OF MATRICES RELIES HEAVILY ON DETERMINANTS,
RANKS, AND TRACES. HORN AND JOHNSON (2012) PROVIDE A THOROUGH EXPLORATION OF
THESE OPERATIONS, OFFERING VALUABLE INSIGHTS INTO THEIR SIGNIFICANCE ACROSS
VARIOUS MATHEMATICAL CONTEXTS. FURTHERMORE, LAY (2012) EXTENSIVELY COVERS
THE STUDY OF INVERSES, SHEDDING LIGHT ON THE SOLVABILITY OF LINEAR SYSTEMS.

EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS ARE ESSENTIAL IN THE EXAMINATION OF LINEAR
TRANSFORMATIONS AND STABILITY ANALYSIS. GOLUB AND VAN LOAN'S INFLUENTIAL
WORK (2013) SERVES AS A DEFINITIVE GUIDE FOR UNDERSTANDING THESE CONCEPTS AND
THEIR APPLICATIONS IN VARIOUS SCIENTIFIC FIELDS. ADDITIONALLY, TREFETHEN AND BAU'S
BOOK (1997) PROVIDES A COMPREHENSIVE EXPLORATION OF NUMERICAL LINEAR
ALGEBRA, WHICH INCLUDES ALGORITHMS FOR COMPUTING EIGENVALUES.

SPECIAL MATRICES AND TRANSPOSITIONS

SPECIAL MATRICES, LIKE ORTHOGONAL MATRICES, HAVE DISTINCT CHARACTERISTICS
THAT FIND USE IN DIVERSE DISCIPLINES. STEWART'S BOOK (1990) PROVIDES A THOROUGH
EXAMINATION OF MATRIX ALGORITHMS, INCLUDING THOSE SPECIFICALLY TAILORED TO
SPECIAL MATRICES. TRANSPOSITIONS, EXTENSIVELY EXPLORED BY MEYER (2000), OFFER
VALUABLE INSIGHTS INTO SYMMETRY AND PATTERN RECOGNITION WITHIN MATRICES.

TOGETHER, THESE WORKS ESTABLISH A STRONG GROUNDWORK FOR COMPREHENDING

THE COMPLEXITIES OF MATRICES, THEIR OPERATIONS, AND THEIR PRACTICAL
APPLICATIONS. THEY SERVE AS INDISPENSABLE RESOURCES FOR ENGINEERS,
MATHEMATICIANS, AND SCIENTISTS WORKING IN FIELDS WHERE LINEAR ALGEBRA HOLDS
SIGNIFICANT IMPORTANCE.

III. MATERIAL & EQUIPMENT

SOFTWARE: MATLAB
TO FACILITATE THIS ANALYSIS, WE WILL UTILIZE MATLAB, A ROBUST COMPUTATIONAL
TOOL. MATLAB'S VAST COLLECTION OF FUNCTIONS FOR NUMERICAL COMPUTATIONS AND
DATA VISUALIZATION MAKES IT AN EXCELLENT CHOICE FOR PERFORMING MATRIX
OPERATIONS AND COMPUTATIONS. WE WILL SPECIFICALLY USE THE LATEST VERSION,
MATLAB R2022, TO ENSURE COMPATIBILITY AND TAKE ADVANTAGE OF THE MOST RECENT
FEATURES AVAILABLE.

HARDWARE: PERSONAL COMPUTER

FOR OPTIMAL PERFORMANCE WHEN USING MATLAB, IT IS RECOMMENDED TO HAVE A
STANDARD PERSONAL COMPUTER WITH AT LEAST 8GB OF RAM AND A DUAL-CORE
PROCESSOR. THIS CONFIGURATION ENSURES SMOOTH AND EFFICIENT EXECUTION OF
COMPUTATIONS, PARTICULARLY WHEN WORKING WITH LARGE MATRICES AND COMPLEX
OPERATIONS.

DATASETS: MATRIX A AND MATRIX B

*DRAW THE MATRIX A AND B HERE*
THE MATRICES GIVEN, A AND B, WILL BE USED AS THE MAIN DATASETS FOR ANALYSIS.
MATRIX A IS A 3X3 MATRIX, WHILE MATRIX B IS A 5X5 MATRIX. THESE MATRICES WILL BE
 LOADED INTO MATLAB TO PERFORM THE SPECIFIED OPERATIONS, SUCH AS
DETERMINANTS, RANKS, TRACES, AND VARIOUS TRANSFORMATIONS.

IV. ILLUSTRATION
>> A=[1 10 20;2 5 6;7 8 9]

A=

1 10 20
2 5 6
7 8 9

>> DETERMINANT_A=DET(A);
>> RANK_A=RANK(A);
>> TRACE_A=TRACE(A);
>> SQUARE_A=A*A;
>> TRANSPOSE_A=TRANSPOSE(A);
>> EIGENVALUE_A=EIG(A)
EIGENVALUE_A =
21.5935
-7.4790
0.8855
>> [VX]=EIG(A);
>> ORTHOGONAL_A=ORTH(A);
>> B=[6 4 2 5 1;3 8 6 4 2;7 1 9 3 5;0 2 4 8 7;5 3 1 6 9]

B=

6 4 2 5 1
3 8 6 4 2
7 1 9 3 5
0 2 4 8 7
5 3 1 6 9

>> DETERMINANT_B=DET(B)

DETERMINANT_B = 2.0062E+04

>> RANK_B=RANK(B);
>> TRACE_B=TRACE(B);
>> SQUARE_B=B*B;
>> TRANSPOSE_B=TRANSPOSE(B);
>> INV_B=INV(B)

INV_B =
 0.1030 -0.0513 0.0458 -0.0882 0.0432
-0.0472 0.1508 -0.0674 -0.0660 0.0606
-0.0488 0.0345 0.0887 0.0447 -0.0863
0.1626 -0.0782 -0.0391 0.1472 -0.0935
-0.1445 0.0265 0.0133 -0.0321 0.1389

>> EIGENVALUE_B=EIG(B);
>> [VX]=EIG(B);
>> ORTHOGONAL_B=ORTH(B);
>>

B*INV_B

ANS =

1.0000 -0.0000 0 -0.0000 0

-0.0000 1.0000 0.0000 -0.0000 0.0000
0.0000 -0.0000 1.0000 -0.0000 -0.0000
-0.0000 0.0000 0.0000 1.0000 0.0000
0.0000 -0.0000 -0.0000 -0.0000 1.0000

V. RESULTS & DISCUSSION

1. DOES THE MATRIX HAVE MORE ROWS OR MORE COLUMNS?

MATRIX B, WITH DIMENSIONS OF 5X5, HAS AN EQUAL NUMBER OF ROWS AND
COLUMNS, INDICATING THAT IT IS A SQUARE MATRIX.

2. WHICH ROW OR COLUMN HAS THE LARGEST NUMBERS? THE SMALLEST?

THE FOURTH COLUMN CONTAINS THE LARGEST SUM OF 26, WHILE THE FIRST ROW AND
SECOND COLUMN CONTAINS THE SMALLEST SUM OF 18.

3. DOES THE MATRIX HAVE A DETERMINANT OF ZERO? IF YES, IT'S SPECIAL AND
MIGHT NOT HAVE AN "OPPOSITE"(OR INVERSE)
NO, MATRIX B DOES NOT HAVE A DETERMINANT OF ZERO. THE DETERMINANT OF
MATRIX B IS 20062. THIS INDICATES THAT MATRIX B HAS AN OPPOSITE AND INVERSE.

4. TRY MULTIPLYING THE MATRIX BY ITS INVERSE. DO YOU GET A MATRIX WHERE THE
MAIN DIAGONAL HAS ALL ONES AND THE OTHER NUMBERS ARC ZEROS?
AFTER MULTIPLYING THE MATRIX B WITH ITS INVERSE, THE ANSWER IS A 5X5 MATRIX
WITH THE MAIN DIAGONAL HAS ALL ONES AND THE OTHER NUMBERS ARE ZERO.

5. COMPARE THE ORIGINAL MATRIX AND ITS TRANSPOSE. WHAT IS CHANGED? CAN YOU
SEE A PATTERN OR SYMMETRY?
THE VALUES IN THE INVERSE MATRIX ARE NOTICEABLY SMALLER IN MAGNITUDE
COMPARED TO MATRIX B. THERE IS NO APPARENT PATTERN OR SYMMETRY BETWEEN
MATRIX B AND ITS INVERSE. THE DIAGONAL ELEMENTS OF THE INVERSE MATRIX ARE
SIGNIFICANTLY LARGER THAN THE OFF-DIAGONAL ELEMENTS. THE OFF-DIAGONAL
ELEMENTS DISPLAY A RANGE OF BOTH POSITIVE AND NEGATIVE VALUES, SUGGESTING
 COMPLEX INTERRELATIONSHIPS BETWEEN THE ELEMENTS.

6. WHAT ARE THE EIGENVALUES OF MATRIX A? CAN A MATRIX HAVE

COMPLEX EIGENVALUES?
THE EIGENVALUES OF MATRIX A ARE APPROXIMATELY [21.5935; -7.4790; 0.8855]. IT IS
POSSIBLE FOR A MATRIX TO HAVE COMPLEX EIGENVALUES. THIS HAPPENS WHEN THE
MATRIX REPRESENTS A TRANSFORMATION THAT INVOLVES ROTATION OR STRETCHING
IN A COMPLEX PLANE.

VI. CONCLUSION

IN THIS ANALYSIS, MATLAB WAS USED TO EXPLORE THE PROPERTIES AND OPERATIONS
OF TWO MATRICES, A AND B. MATRIX B, WITH DIMENSIONS OF 5X5, DISPLAYED UNIQUE
CHARACTERISTICS, INCLUDING ITS INVERTIBILITY. THE DETERMINANTS, INVERSES, AND
EIGENVALUES OF MATRIX B WERE COMPUTED, PROVIDING INSIGHTS INTO THE
TRANSFORMATIONS REPRESENTED BY THIS MATRIX.

MATRIX A, ON THE OTHER HAND, IS A 3X3 MATRIX THAT EXHIBITED DIFFERENT

PROPERTIES, SUCH AS ITS INVERTIBILITY. THE DETERMINANTS, INVERSES, AND
EIGENVALUES OF MATRIX A WERE CALCULATED, SHOWCASING ITS DISTINCTIVE
MATHEMATICAL ATTRIBUTES.

AFTER CONDUCTING A THOROUGH EXAMINATION, WE COMPARED MATRIX B WITH ITS

INVERSE. THE INVERSE OF MATRIX B DISPLAYED SIGNIFICANTLY SMALLER VALUES IN
MAGNITUDE COMPARED TO THE ORIGINAL MATRIX. ALTHOUGH NO DISTINCT PATTERN OR
SYMMETRY EMERGED, THE DIAGONAL ELEMENTS OF THE INVERSE WERE NOTICEABLY
LARGER THAN THE OFF-DIAGONAL ELEMENTS, HIGHLIGHTING THEIR IMPORTANCE IN
MATRIX BEHAVIOR. THE OFF-DIAGONAL ELEMENTS EXHIBITED A COMBINATION OF
POSITIVE AND NEGATIVE VALUES, INDICATING COMPLEX INTERRELATIONSHIPS WITHIN THE
MATRIX.

IN CONCLUSION, THIS COMPREHENSIVE ANALYSIS HAS PROVIDED US WITH A DEEPER

UNDERSTANDING OF THE PROPERTIES, OPERATIONS, AND IMPLICATIONS OF MATRICES A
AND B. THIS KNOWLEDGE IS VALUABLE NOT ONLY IN THE FIELD OF LINEAR ALGEBRA BUT
ALSO IN ITS DIRECT APPLICATIONS IN ENGINEERING AND MATHEMATICS. IT CONTRIBUTES
SIGNIFICANTLY TO ADVANCING OUR UNDERSTANDING IN THESE DISCIPLINES.

VII. RECOMMENDATION

EMBARK ON A COMPREHENSIVE EXPLORATION OF SQUARE MATRICES, FOCUSING ON

THEIR UNIQUE PROPERTIES, PARTICULARLY THOSE THAT EXHIBIT SPECIAL CHARACTERISTICS
LIKE ORTHOGONAL MATRICES. THIS IN-DEPTH STUDY WILL EXPAND YOUR UNDERSTANDING
OF LINEAR ALGEBRA, EQUIPPING YOU WITH THE NECESSARY TOOLS TO APPROACH A WIDE
RANGE OF APPLICATIONS. MOREOVER, DELVING INTO ADVANCED CONCEPTS SUCH AS
EIGENVALUE DECOMPOSITION WILL PROVIDE VALUABLE INSIGHTS FOR TACKLING COMPLEX
ENGINEERING PROBLEMS AND CONDUCTING RIGOROUS ANALYSES.

EXPLORE MORE ADVANCED TOPICS IN LINEAR ALGEBRA, SUCH AS EIGENVALUE

 DECOMPOSITION AND SINGULAR VALUE DECOMPOSITION. THESE TOOLS ARE CRUCIAL FOR
DISSECTING EVEN THE MOST INTRICATE TRANSFORMATIONS. THIS KNOWLEDGE IS
PARTICULARLY ADVANTAGEOUS IN SCENARIOS WHERE A DEEP UNDERSTANDING OF THE
UNDERLYING MATHEMATICAL PRINCIPLES IS CRUCIAL. IT EMPOWERS YOU TO CONFIDENTLY
HANDLE COMPLEX SYSTEMS, MAKING IT AN INVALUABLE ASSET IN BOTH YOUR ACADEMIC
PURSUITS AND PROFESSIONAL ENDEAVORS.

VIII. REFERENCES

GILAT, A. (2014). MATLAB: AN INTRODUCTION WITH APPLICATIONS (5TH ED.). WILEY.

GOLUB, G. H., & VAN LOAN, C. F. (2013). MATRIX COMPUTATIONS (4TH ED.). JOHNS
HOPKINS UNIVERSITY PRESS.

HORN, R. A., & JOHNSON, C. R. (2012). MATRIX ANALYSIS (2ND ED.). CAMBRIDGE

UNIVERSITY PRESS. LAY, D. C. (2012). LINEAR ALGEBRA AND ITS APPLICATIONS (4TH ED.).

PEARSON.

LUENBERGER, D. G. (2008). LINEAR AND NONLINEAR PROGRAMMING (3RD ED.).

SPRINGER. MEYER, C. D. (2000). MATRIX ANALYSIS AND APPLIED LINEAR ALGEBRA.

SIAM.

STEWART, G. W. (1990). MATRIX ALGORITHMS, VOLUME I: BASIC DECOMPOSITIONS. SIAM.

STRANG, G. (2005). INTRODUCTION TO LINEAR ALGEBRA (4TH ED.). WELLESLEY-CAMBRIDGE

PRESS. TREFETHEN, L. N., & BAU, D. III. (1997). NUMERICAL LINEAR ALGEBRA. SIAM.