1. Matrix (mathematics) - Wikipedia, The Free Encyclopedia In mathematics, a matrix (plural matrices) is a rectangular table of elements . Matrix multiplication is not commutative; that is, given matrices A and B http://en.wikipedia.org/wiki/Matrix_(mathematics)

Matrix (mathematics) From Wikipedia, the free encyclopedia Jump to: navigation search For the square matrix section, see square matrix In mathematics , a matrix (plural matrices ) is a rectangular table of elements (or entries ), which may be numbers or, more generally, any abstract quantities that can be added and multiplied . Matrices are used to describe linear equations , keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices are described by the field of matrix theory . Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra In this article, the entries of a matrix are real or complex numbers unless otherwise noted. Organization of a matrix Contents Definitions and notations Mathematical definition Example Adding and multiplying matrices ... edit Definitions and notations The horizontal lines in a matrix are called rows and the vertical lines are called columns . A matrix with m rows and n columns is called an m -by- n matrix (written m n ) and m and n are called its dimensions . The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an

2. QuickMath Automatic Math Solutions The matrices section of QuickMath allows you to perform arithmetic operations on For instance, when adding two matrices A and B, the element at row i, http://www.quickmath.com/www02/pages/modules/matrices/index.shtml

Algebra Expand Factor Simplify ... Determinant Graphs Equations Inequalities Numbers Percentages Scientific notation google_ad_client = "pub-8651647546713104"; google_ad_width = 120; google_ad_height = 600; google_ad_format = "120x600_as"; google_ad_type = "text"; google_ad_channel ="7739705934"; google_color_border = "336699"; google_color_bg = "FFFFFF"; google_color_link = "0000FF"; google_color_url = "008000"; google_color_text = "000000"; Matrices The matrices section of QuickMath allows you to perform arithmetic operations on matrices. Currently you can add or subtract matrices, multiply two matrices, multiply a matrix by a scalar and raise a matrix to any power. What is a matrix? A matrix is a rectangular array of elements (usually called scalars), which are set out in rows and columns. They have many uses in mathematics, including the transformation of coordinates and the solution of linear systems of equations. Here is an example of a 2x3 matrix : Arithmetic The arithmetic suite of commands allows you to add or subtract matrices, carry out matrix multiplication and scalar multiplication and raise a matrix to any power. Matrices are added to and subtracted from one another element by element. For instance, when adding two matrices A and B, the element at row i, column j of A is added to the element at row i, column j of B to give the element at row i, column j of the answer. Consequently, you can only add and subtract matrices which are the same size.

3. Algebra II: Matrices - Math For Morons Like Us On this page we hope to clear up problems that you might have with matrices. matrices are good things to have under control and know how to deal with, http://library.thinkquest.org/20991/alg2/matrices.html

Systems of Eq. Polynomials Frac. Express. Complex Numbers ... Trig. Identities On this page we hope to clear up problems that you might have with matrices. Matrices are good things to have under control and know how to deal with, because you will use them extensively in pre-calculus to solve systems of equations that have variables up the wazoo! (Like one we remember with seven equations in seven variables.) Addition and subtraction Multiplication Quiz on Matrices To add matrices, we add the corresponding members. The matrices have to have the same dimensions. Example: Solution: Add the corresponding members. Subtraction of matrices is done in the same manner as addition. Always be aware of the negative signs and remember that a double negative is a positive! Back to Top You can multiply a matrix by another matrix or by a number. When you multiply a matrix by a number, multiply each member of the matrix by the number. To multiply a matrix by a matrix, the first matrix has to have the same number of columns as the rows in the second matrix. Examples: Solution: Multiply each member of the matrix by 2. Problem: Multiply the matrices shown below.

4. AMS Online Books/Letters On Matrices/COLL17 The 1934 classic Lectures on matrices by Wedderburn in scanned PDF. http://www.ams.org/online_bks/coll17/

Title List Help AMS Home AMS Bookstore Lectures on Matrices by J. H. M. Wedderburn Publication Date: 1934 Number of Pages: 205pp. Publisher: AMS ISBN:0-8218-3204-2 COLL17.E Download Individual Chapters FREE (12 files - 13mb) Frontmatter Title Preface Contents Corrigenda Matrices and Vectors Algebraic Operations with Matrices. The Characteristic Equation Invariant Factors and Elementary Divisors Vector Polynomials. Singular Matric Polynomials ... Endmatter Appendix I Notes Appendix II Bibliography Index to Bibliography Index Comments: webmaster@ams.org Privacy Statement Search the AMS

5. S.O.S. Math - Matrix Algebra Introduction to Determinants Determinants of matrices of Higher Order Determinant and Inverse of matrices Application of Determinant to Systems http://www.sosmath.com/matrix/matrix.html

S.O.S. Homepage Algebra Trigonometry Calculus ... CyberBoard Search our site! S.O.S. Math on CD Sale! Only $19.95. Works for PCs, Macs and Linux. Tell a Friend about S.O.S. Books We Like Math Sites on the WWW S.O.S. Math Awards ... Matrix Exponential Applications: Application of Invertible Matrices: Coding Complex numbers as Matrices Markov Chains Systems of Linear Equations System of Equations: An Introduction Systems of Linear Equations: Gaussian Elimination Systems of linear equations in Two variables. Systems of linear equations in Three variables. ... More Problems on Linear Systems and Matrices Determinants Introduction to Determinants Determinants of Matrices of Higher Order Determinant and Inverse of Matrices Application of Determinant to Systems: Cramer's Rule Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors: An Introduction Computation of Eigenvalues Computation of Eigenvectors The Case of Complex Eigenvalues ... Diagonalization APPENDIX Tables Contact us Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA users online during the last hour

6. Matrices And Determinants The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants

Matrices and determinants Algebra index History Topics Index Version for printing The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17 th Century that the ideas reappeared and development really got underway. It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains the following problem:- There are two fields whose total area is square yards. One produces grain at the rate of of a bushel per square yard while the other produces grain at the rate of a bushel per square yard. If the total yield is bushels, what is the size of each field. The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:-

7. Matrices And Determinants DEFINITION Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html

On this page will be: Introduction and Examples Matrix Addition and Subtraction Matrix Multiplication The Transpose of a Matrix ... Cramer's Rule Introduction and Examples DEFINITION: A matrix is defined as an ordered rectangular array of numbers. They can be used to represent systems of linear equations, as will be explained below Here are a couple of examples of different types of matrices: Symmetric Diagonal Upper Triangular Lower Triangular Zero Identity And a fully expanded mxn matrix A, would look like this: or in a more compact form: Matrix Addition and Subtraction DEFINITION: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of rows and columns. Take: Addition If A and B above are matrices of the same type then the sum is found by adding the corresponding elements a ij b ij Here is an example of adding A and B together Subtraction If A and B are matrices of the same type then the subtraction is found by subtracting the corresponding elements a ij b ij Here is an example of subtracting matrices Now, try adding and subtracting your own matrices

8. Matrices Worksheets, Determinants, Cramer's Rule, And More. Determinants Mix of 2 x 2 and 3 x 3 matrices Determinants Calculate area of triangles Augmented matrices Write the Augmented Matrix and Solve http://edhelper.com/Matrices.htm

Return to edHelper.com Matrices Worksheets Also Visit: Algebra Worksheets Matrices Worksheets Addition of Matrices Subtraction of Matrices Multiply a Matrix by One Number Addition and Subtraction ... Final Review of Matrices

9. An Introduction To MATRICES r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above. http://home.scarlet.be/~ping1339/matr.htm

An introduction to MATRICES Definitions Matrix Square matrix Diagonal matrix ... Theorem 5 Definitions Matrix A matrix is an ordered set of numbers listed rectangular form. Example. Let A denote the matrix This matrix A has three rows and four columns. We say it is a 3 x 4 matrix. We denote the element on the second row and fourth column with a Square matrix If a matrix A has n rows and n columns then we say it's a square matrix. In a square matrix the elements a i,i , with i = 1,2,3,... , are called diagonal elements. Remark. There is no difference between a 1 x 1 matrix and an ordenary number. Diagonal matrix A diagonal matrix is a square matrix with all de non-diagonal elements 0. The diagonal matrix is completely denoted by the diagonal elements. Example. [7 0] [0 5 0] [0 6] The matrix is denoted by diag(7 , 5 , 6) Row matrix A matrix with one row is called a row matrix Column matrix A matrix with one column is called a column matrix Matrices of the same kind Matrix A and B are of the same kind if and only if A has as many rows as B and A has as many columns as B The tranpose of a matrix The n x m matrix A' is the transpose of the m x n matrix A if and only if The ith row of A = the ith column of A' for (i = 1,2,3,..n)

10. Rotation Matrix -- From Wolfram MathWorld Orthogonal matrices have special properties which allow them to be manipulated be two orthogonal matrices. By the orthogonality condition, they satisfy http://mathworld.wolfram.com/RotationMatrix.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Less... Rotation Matrix When discussing a rotation , there are two possible conventions: rotation of the axes , and rotation of the object relative to fixed axes. In , consider the matrix that rotates a given vector by a counterclockwise angle in a fixed coordinate system. Then so This is the convention used by Mathematica command RotationMatrix theta On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle . The coordinates of the fixed vector in the rotated coordinate system are now given by a rotation matrix which is the matrix transpose of the fixed-axis matrix and, as can be seen in the above diagram, is equivalent to rotating the vector by a counterclockwise angle of relative to a fixed set of axes, giving This is the convention commonly used in textbooks such as Arfken (1985, p. 195). In , coordinate system rotations of the x y -, and z -axes in a counterclockwise direction when looking towards the origin give the matrices (Goldstein 1980, pp. 146-147 and 608; Arfken 1985, pp. 199-200).

11. The History Of Matrices The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200 http://ualr.edu/lasmoller/matrices.html

Did you know . . .? The history of matrices goes back to ancient times! But the term "matrix" was not applied to the concept until 1850. "Matrix" is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced. The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art Chiu Chang Suan Shu ), gives the first known example of the use of matrix methods to solve simultaneous equations. In the treatise's seventh chapter, "Too much and not enough," the concept of a determinant first appears, nearly two millennia before its supposed invention by the Japanese mathematician Seki Kowa in 1683 or his German contemporary Gottfried Leibnitz (who is also credited with the invention of differential calculus, separately from but simultaneously with Isaac Newton). More uses of matrix-like arrangements of numbers appear in chapter eight, "Methods of rectangular arrays," in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by

12. Tim Davis: UF Sparse Matrix Collection : Sparse Matrices From A Wide Range Of Ap A collection of large sparse matrices from many scientific disciplines with links and software pieces to operate on matrix data structures. http://www.cise.ufl.edu/research/sparse/matrices/

University of Florida Sparse Matrix Collection: Maintained by Tim Davis From the abstract of the paper The University of Florida Sparse Matrix Collection: As of September 2007, it contains 1877 problems (some of which are sequences of dozens of matrices). The smallest is 5-by-5 with 19 nonzero entries. The largest has dimension 9.8 million, and the matrix with the most nonzeros has 99.2 million entries. The matrices are available in three formats: MATLAB mat-file, Rutherford-Boeing, and Matrix Market. The size of the collection in each format is about 9 GB. Note that the MATLAB mat-files can only be read by MATLAB 7.0 or later. This collection is managed by Tim Davis, but ``editors'' of other collections are attributed, via the Problem.ed field in each problem set. Problem.author is the matrix creator. Other collections are always welcome. Click here for a paper describing the collection (Jan. 2007). Note: all of the matrices have been updated as of November 25, 2006. Additional minor changes to the meta-data were made in January, 2007 (problem kind added to all matrices). Most changes are minor, but if you have existing matrices from this collection, I suggest you delete them and download the most recent copies.

13. BLAST Substitution Matrices The theory of amino acid substitution matrices is described in 1, and applied to DNA sequence comparison in 2. In general, different substitution http://www.ncbi.nlm.nih.gov/blast/html/sub_matrix.html

BLAST substitution matrices Query Length Substitution Matrix Gap Costs PAM-30 PAM-70 BLOSUM-80 BLOSUM-62 Gap Costs The raw score of an alignment is the sum of the scores for aligning pairs of residues and the scores for gaps. Gapped BLAST and PSI-BLAST use "affine gap costs" which charge the score -a for the existence of a gap, and the score -b for each residue in the gap. Thus a gap of k residues receives a total score of -(a+bk); specifically, a gap of length 1 receives the score -(a+b). Lambda Ratio To convert a raw score S into a normalized score S' expressed in bits, one uses the formula S' = (lambda*S - ln K)/(ln 2), where lambda and K are parameters dependent upon the scoring system (substitution matrix and gap costs) employed [7-9]. For determining S', the more important of these parameters is lambda. The "lambda ratio" quoted here is the ratio of the lambda for the given scoring system to that for one using the same substitution scores, but with infinite gap costs [8]. This ratio indicates what proportion of information in an ungapped alignment must be sacrificed in the hope of improving its score through extension using gaps. We have found empirically that the most effective gap costs tend to be those with lambda ratios in the range 0.8 to 0.9. Altschul, S.F. (1993) "A protein alignment scoring system sensitive at all evolutionary distances." J. Mol. Evol. 36:290-300.

14. Hadamard Matrices A library of Hadamard matrices maintained by NJA Sloane. http://www.research.att.com/~njas/hadamard/

A Library of Hadamard Matrices N. J. A. Sloane Keywords : Hadamard matrices, Kimura matrices Paley matrices, Plackett-Burman designs, Sylvester matrices, Turyn construction, Williamson construction Contains all Hadamard matrices of orders n up through 28, and at least one of every order n up through 256. This library is maintained by N. J. A. Sloane njas@research.att.com Notation: had.n.name indicates a Hadamard matrix of order n and type "name". The matrices are usually given as n rows each containing n +'s and -'s (with no spaces). In many cases there are further rows giving the name of the matrix and the order of its automorphism group. What the suffixes mean: od = orthogonal design construction method pal = first Paley type pal2 = second Paley type syl = Sylvester type tur = Turyn type tx = tensor product of type x with ++/+- or (rarely) with a Hadamard matrix of order 4 will = Williamson type References: Seberry, J. and Yamada, M., Hadamard matrices, sequences, and block designs , pp. 431-560 of Dinitz, J. H. and Stinson, D. R., editors (1992), Contemporary Design Theory: A Collection of Essays, Wiley, New York. Chapter 7 of Orthogonal Arrays by Hedayat, Sloane and Stufken.

15. Homogeneous Transformation Matrices Explicit ndimensional homogeneous matrices for projection, dilation, reflection, shear, strain, rotation and other familiar transformations. http://www.silcom.com/~barnowl/HTransf.htm

HOMOGENEOUS TRANSFORMATION MATRICES Daniel W. VanArsdale Vector (nonhomogeneous) methods are still being recommended to effect rotations and other linear transformations. Homogeneous matrices have the following advantages: simple explicit expressions exist for many familiar transformations including rotation these expressions are n-dimensional there is no need for auxiliary transformations, as in vector methods for rotation more general transformations can be represented (e.g. projections, translations) directions (ideal points) can be used as parameters of the transformation, or as inputs if nonsingular matrix T transforms point P by PT, then hyperplane h is transformed by T h the columns of T (as hyperplanes) generate the null space of T by intersections many homogeneous transformation matrices display the duality between invariant axes and centers. The expressions below use reduction to echelon form and Gram-Schmidt orthonormalization, both with slight modifications. They can be easily coded in any higher level language so that the same procedures generate transformations for any dimension. This article is at an undergraduate level, but the reader should have had some exposure to linear algebra and analytic projective geometry. This material is based on: Daniel VanArsdale, Homogeneous Transformation Matrices for Computer Graphics, , vol. 18, no. 2, pp. 177-191, 1994. Some

16. Substitution Matrices In aligning two protein sequences, some method must be used to score the alignment of one residue against another. Substitution matrices contain such values http://arep.med.harvard.edu/seqanal/submatrix.html

Substitution Matrices In aligning two protein sequences, some method must be used to score the alignment of one residue against another. Substitution matrices contain such values. Widely used matrices PAM / MDM / Dayhoff The late Margaret Dayhoff was a pioneer in protein databasing and comparison. She and her coworkers developed a model of protein evolution which resulted in the development of a set of widely used substitution matrices . These are frequently called Dayhoff, MDM (Mutation Data Matrix), or PAM (Percent Accepted Mutation) matrices. Derived from global alignments of closely related sequences. Matrices for greater evolutionary distances are extrapolated from those for lesser ones. The number with the matrix (PAM40, PAM100) refers to the evolutionary distance; greater numbers are greater distances. Several later groups have attempted to extend Dayhoff's methodology or re-apply her analysis using later databases with more examples. Extensions Jones, Thornton and coworkers used the same methodology as Dayhoff but with modern databases ( CABIOS 8:275 Gonnett and coworkers ( Science 256:1443 ) used a slightly different (but theoretically equivalent) methodology Proteins 17:49 ) compared these two newer versions of the PAM matrices with Dayhoff's originals.

17. Raven Standard Progressive Matrices The Standard Progressive matrices (SPM) was designed to measure a person’s ability to form perceptual relations and to reason by analogy...... http://www.cps.nova.edu/~cpphelp/RSPM.html

Raven Standard Progressive Matrices Purpose: Designed to measure a person’s ability to form perceptual relations. Population: Ages 6 to adult. Score: Percentile ranks. Time: (45) minutes. Author: J.C. Raven. Publisher: U.S. Distributor: The Psychological Corporation. Description: The Standard Progressive Matrices (SPM) was designed to measure a person’s ability to form perceptual relations and to reason by analogy independent of language and formal schooling, and may be used with persons ranging in age from 6 years to adult. It is the first and most widely used of three instruments known as the Raven's Progressive Matrices, the other two being the Coloured Progressive Matrices (CPM) and the Advanced Progressive Matrices (APM). All three tests are measures of Spearman's g. Scoring: Reliability: Internal consistency studies using either the split-half method corrected for length or KR20 estimates result in values ranging from .60 to .98, with a median of .90. Test-retest correlations range from a low of .46 for an eleven-year interval to a high of .97 for a two-day interval. The median test-retest value is approximately .82. Coefficients close to this median value have been obtained with time intervals of a week to several weeks, with longer intervals associated with smaller values. Raven provided test-retest coefficients for several age groups: .88 (13 yrs. plus), .93 (under 30 yrs.), .88 (30-39 yrs.), .87 (40-49 yrs.), .83 (50 yrs. and over). Validity: Spearman considered the SPM to be the best measure of g. When evaluated by factor analytic methods which were used to define g initially, the SPM comes as close to measuring it as one might expect. The majority of studies which have factor analyzed the SPM along with other cognitive measures in Western cultures report loadings higher than .75 on a general factor. Concurrent validity coefficients between the SPM and the Stanford-Binet and Weschler scales range between .54 and .88, with the majority in the .70s and .80s.

18. Science News Online - Ivars Peterson's MathLand - 6/14/97 Illustrated article explaining how contra dance patterns and rhythms are formed. http://www.sciencenews.org/sn_arc97/6_14_97/mathland.htm

June 14, 1997 Contra Dancing and Matrices The origins of contra dancing go back to colonial days, and its roots can be traced to English country dance. It's really a group rather than a couples effort, and it has elements that might remind you of traditional square dancing. Rhythm and pattern are the keys. What's striking, says Scanlon, is that a remarkably high percentage of its practitioners are highly educated, often involved in mathematics, computers, or engineering. "The appeal seems to lie in its being a kind of 'set dancing,' where one's position relative to others while tracing patterns on the dance floor is paramount," he says. "Timing is also crucial, as is the ability to rapidly carry out called instructions and do fraction math on the fly." Scanlon introduced both the mathematical and performance sides of contra dancing to attendees earlier this year at the 2nd Annual Recreational Mathematics Conference (see "Fun and Games in Nevada" The music for contra dancing is highly structured. Everything occurs in units of four. The band plays a tune for 16 beats, repeats the tune, then plays a new tune for 16 beats and repeats that. An eight-beat section is known as a call, during which each block of four dancers executes a called-out instruction. An entire dance is precisely 64 beats long. When the dancers line up in their groups of four to produce a long column down the floor extending away from the band, each square block consisting of two couples can be thought of as a matrix. Each dancer (element of the matrix) is in a specific position within the block. The called instructions correspond to rearrangements of the elements of the matrix. After 64 beats, however, the first and second rows of the matrix must be interchanged. Of course, that can be done in one step, but the fun comes in all the different ways in which groups of four can get to that inevitable end result.

19. Toeplitz And Circulant Matrices A very old (1971, revised 1977, 1993, 1997, 1998, 2000, 2001, 2002, 2005, 2006) but still occasionally useful tutorial on Toeplitz and circulant matrices. http://www-ee.stanford.edu/~gray/toeplitz.html

T oeplitz and Circulant Matrices Toeplitz and Circulant Matrices: A Review , by R. M. Gray. A very old (1971, revised 1977, 1993, 1997, 1998, 2000, 2001, 2002, 2005, 2006) but still occasionally useful tutorial on Toeplitz and circulant matrices. The report was revised with the help of two very thorough reviewers and is being published both online and as a paperback book by NOW publishers. The official citation to the published version is R. M. Gray, "Toeplitz and Circulant Matrices: A review" Foundations and Trends in Communications and Information Theory, Vol 2, Issue 3, pp 155-239, 2006. Journal reprint Note: The typos found and noted below are corrected in the first pdf, but not in the second. A printed and bound version of the paperback book is available at a 35% discount from Now Publishers. This can be obtained by entering the promotional code on the order form at now publishers. You will then pay only $28.00 including postage. (The Website is due to be activated as soon as the book is available.) Typographical errors: On p. (62) the explanation of (5.1) is garbled. It published latex is

20. Mathematics Reference: Rules For Matrices Basic properties of matrices. A, B, and C are matrices,; O represents the zero matrix,; I represents the identity matrix,; r, s, and n are scalars. http://www.alcyone.com/max/reference/maths/matrices.html

Mathematics reference Rules for matrices Ma MathRef Basic properties of matrices. Legend. A, B, and C are matrices, O represents the zero matrix, I represents the identity matrix, r s , and n are scalars. Basic. -A == (-1) A equation 1 A - B == A + (-B) equation 2 1 A = A equation 3 A = O equation 4 A + O = O + A = A equation 5 I A = A I = A equation 6 A - A = O equation 7 Addition and scalar product. A + B = B + A equation 8 (A + B) + C = A + (B + C) equation 9 r (A + B) = r A + r B equation 10 r s ) A = r A + s A equation 11 r s ) A = r s A) equation 12 Matrix product. A == I equation 13 A == A A equation 14 A n = A A n equation 15 (A B) C = A (B C) equation 16 A (B + C) = A B + A C equation 17 (A + B) C = A C + B C equation 18 Transpose and inverse. I T = I equation 19 (A T T = A equation 20 (A + B) T = A T + B T equation 21 r A) T r A T equation 22 (A B) T = B T A T equation 23 I = I equation 24 A A = A A = I equation 25 (A B) = B A equation 26 (A T = (A T equation 27 Trace. tr (A + B) = tr A + tr B equation 28 tr ( r A) = r tr A equation 29 tr (A B) = tr (B A) equation 30 Determinant and adjoint.

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