Text. 7. Summary. A. Gilat, MATLAB: An Introduction with Applications, 4th ed., Wiley Wikipedia: Matrix, SLE, etc.

Transcription

1 2. Linear algebra 1. Matrix operations 2. Determinant and matrix inverse 3. Advanced matrix manipulations in the MATLAB 4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination 5. Systems of linear equations with the tridiagonal matrix. The Thomas algorithm (optional) 6. Solution of SLE in the MATLAB. Problems 7. Summary Text A. Gilat, MATLAB: An Introduction with Applications, 4th ed., Wiley Wikipedia: Matrix, SLE, etc. ME 349, Engineering Analysis, Alexey Volkov 1

2 2.1. Matrix operations Motivation: Solution of SLE Notion of a matrix Productofamatrixandanumber,sumofmatrix,matrix transpose, product of matrices Special matrices Reading assignment Gilat, 3.1, 3.2, 3.4, and 3.5 ME 349, Engineering Analysis, Alexey Volkov 2

3 2.1. Matrix operation Motivation: Solution of systems of linear equation The linear function is the function of the form, where,. The linear equation is the equation of the form = 0. Solution (root) /. The system of linear equations (SLE) of equations with unknowns is the system : Row index = index of equation, Column index = index of unknown, coefficients of equations (1,..,, 1,..,)., unknowns (1,..,)., right hand sides (RHSs) of equations (1,..,). To solve a SLE means to find such that turn every equation in the system into identity. ME 349, Engineering Analysis, Alexey Volkov 3

4 2.1. Matrix operation Applications of SLEs: Numerical solutions of partial differential equations: Computational heat transfer, computational fluid mechanics, computational solid mechanics, and many more... ME 349, Engineering Analysis, Alexey Volkov 4

5 2.1. Matrix operations In mathematics, the matrix is a (two dimensional) table of values. Example: Matrix of 5 x 60 values. T(1,1) T(1,2)... T(1,j 1) Column j T(1,j) T(1,j+1)... T(1,59) T(1,60) Row i... T(i,1) T(i,2)... T(i,j 1) T(i,j) T(i,j+1)... T(i,59) T(i,60)... T(5,1) T(5,2)... T(5,j 1) T(5,j) T(5,j+1)... T(5,59) T(5,60) Every entry in the table of the matrix is called the element of the matrix. Every element of the matrix is identified by two integers: The row index and column index. The size of the matrix is the number of rows and number of columns :. Mathematical notation:,, :Matrix of size,,,,,, :Element of matrix in row and column ME 349, Engineering Analysis, Alexey Volkov 5

6 2.1. Matrix operations Basic operations with matrices Scalar multiplication. Product of number and matrix of size is the matrix of size,where Can be calculated for arbitrary and. Matrix addition: Sum of two matrixes and B of size is the matrix of the same size,where Can be calculated only if both matrices and have the same size. We can define subtraction: Difference of two matrices is 1. Matrix transpose: Transpose of matrix of size is the matrix of size,where Examples:,, ME 349, Engineering Analysis, Alexey Volkov 6

7 2.1. Matrix operations Matrix multiplication: Product of two matrices and of sizes and is the matrix of size,where Can be calculated only if, i.e. if the number of columns in is equal to the number of rows in. ( or may not exist: :, : 1, exists, does not exist). Examples:,, Properties: Can be proved by direct evaluation of the left and right hand sides (LHSs and RHSs),,,, ME 349, Engineering Analysis, Alexey Volkov 7

8 2.1. Matrix operations Special matrices Let be a matrix of size. is the row vector if the number of rows is equal to 1 (1):,. is the column vector if the number of columns is equal to 1 (1):,. Transpose of a column vector is a row vector. The zero matrix is the matrix with all zero elements. is the square matrix if the number of rows is equal to the number of columns (). The main diagonal of the square matrix is the diagonal where. Square matrix is called symmetric if. Consequence:. Square matrix is called skew symmetric if. Consequence: 0,. Square matrix is called the identity matrix if 1, 0for. Properties: Examples:,,, is the square matrix,, ME 349, Engineering Analysis, Alexey Volkov 8, , Row Column Symmetric Skew symmetric Identity Zero

9 2.2. Determinant and matrix inverse Determinant Matrix inverse Determinant and matrix inverse in the MATLAB Cramer's rule for matrix inverse Reading assignment Gilat, 3.5 ME 349, Engineering Analysis, Alexey Volkov 9

10 2.2. Determinant and matrix inverse Determinant The determinant det of a square matrix is a uniquely defined numerical value that is calculated based on the elements of the matrix with special rules. Notation:, det det There are a lot of ways to introduce the rules that determine the values of the determinant. We will consider not the best, but simplest iterative rule of introducing the determinant. 1. For every element of matrix of size, let's introduce a square submatrix of size 1 1, which is obtained from by removing all elements in row and column.the(first) minor of corresponding to is det. 2. For every element of matrix, let's introduce the signature that is equal to 1 1 if the number of "steps" from along rows and columns is even 1 otherwise number of "steps" is odd 3. The cofactor of is. Example:, 1 1 1, 1 1 1, ME 349, Engineering Analysis, Alexey Volkov 10

11 2.2. Determinant and matrix inverse 3. Determinant of the square matrix of size 1 x 1 is its element: det. 4. The determinant of the square matrix of arbitrary size can be calculated as: det Eq. (2.2.1) reduces calculation of the determinant of a matrix of size to determinants of matrices of smaller size 1 1. Now we can apply the same rule for and reduces to determinants of matrices of size 2 2 etc. until we get matrixes of size 1 x 1, i.e. just numbers. Eq. (2.2.1) gives us the determinant in the form of "expansion" of matrix along its 1 st row. It can be proved (but we will not do it) that we can obtain the same value det if we will calculate the determinant in the form of "expansion" along any row or column of the matrix : ME 349, Engineering Analysis, Alexey Volkov 11 det det (2.2.1) (2.2.2) (2.2.3) The drawback of our definition of det, Eq. (2.2.1), is that it is difficult to prove (2.2.2) and (2.2.3).

12 2.2. Determinant and matrix inverse Examples:Determinantsofmatrices2x2and3x3. "Expansion" along the first row "Expansion" along the second column Some properties of the determinant (without prove): det det,det 0. det det det. Let's consider a matrix that is obtained from by swapping of any pair of rows or columns. Then det det. Let's consider a matrix which is obtained from by multiplying any single column or row in by constant.thendet det. In particular, det det. Addition to a column or row of any other column or row multiplied by constant does not change det. ME 349, Engineering Analysis, Alexey Volkov 12

13 2.2. Determinant and matrix inverse Let's consider an upper or low triangular matrix : 0 or E.g., for upper triangular matrix: det Determinant of any upper or low triangular matrix is equal to the product of elements on the main diagonal. For diagonal matrix: det For identity matrix: det det In order to simplify calculations of a determinant by hand, we need to choose a row or column with maximum number of zeros. ME 349, Engineering Analysis, Alexey Volkov 13

14 2.2. Determinant and matrix inverse Calculation of det for large is extremely lengthy procedure. It requires ~! individual arithmetic operations: 10, ~ ; 100, ~ For practical calculations of determinants at large, thedecomposition methods are used. They require ~ of arithmetic operations: 10, ~10 ; 100, ~ 10. Geometrical meaning of determinants of 2 x 2 and 3 x 3 matrices (without prove),,,,,,,, det = signed (±) area of parallelogram det = signed (±) volume of parallelepiped ME 349, Engineering Analysis, Alexey Volkov 14

15 2.2. Determinant and matrix inverse Matrix inverse Let's consider a square matrix of size. Thematrix of size is called the matrix inverse to or the inverse matrix if (2.2.4) Existence of the inverse matrix: Let's apply the properties of the determinant: det det det det 1 Thus, The matrix is invertible (has the inverse) only if det 0. The matrix is called singular or degenerate if det 0. If the matrix is invertible, then det 1 det (2.2.5) ME 349, Engineering Analysis, Alexey Volkov 15

16 2.2. Determinant and matrix inverse Cramer's rule (without prove) Cramer's rule says that the inverse of any invertible square matrix can be found in the form:, det 0, then where are cofactors of elements of matrix. According to Cramer's rule, we need to perform three steps in order to find : 1. To calculate cofactors for all elements of and form the matrix 2. To find the transpose of : 3. To calculate determinant det and divide by det : 1 det ME 349, Engineering Analysis, Alexey Volkov 16 (2.2.5)

17 2.2. Determinant and matrix inverse Example: Matrix 2 x , det 0 det 1 Finding the matrix inverse with Cramer's rule is a lengthy operation. It requires calculation of of determinants of size 1 1. If the determinants are calculated according to the expansion formula, Eq. (2.2.1), then the total number of arithmetic operations ~ 1!!. In practical calculations, the decomposition methods are used to calculate minors, and the time complexity of Cramer's rule drops down to ~ and even to ~. ME 349, Engineering Analysis, Alexey Volkov 17

18 2.3. Advanced matrix manipulations in MATLAB Matrix operation in the MATLAB Examples Reading assignment Gilat, 3.1, 3.2, 3.4, and 3.5 ME 349, Engineering Analysis, Alexey Volkov 18

19 2.3. Advanced matrix manipulations in MATLAB Matrix operations in the MATLAB Matrix isa2darray:a=[12;34]; Column vector is a 2D array where the number of columns is equal to 1: a = [ 1 ; 2 ] ; Row vector is a 2D array, where the number of rows is equal to 1: a = [ 1 4 ] ; Function size ( A ) returns the numbers of rows and columns of the matrix A. Matrix operations: Scalar multiplication:a=2;b=[12;34];c=a*b; Addition (subtraction): d=[ 12; 34];f=d+b; Transpose:g=[123;345];h=g'; Matrix multiplication:w=b*g; Matrix power (only for square matrices): p = b^3 is equivalent to p = b * b * b. Function det ( A ) returns the determinant of the square matrix A. Function inv ( A ) returns the inverse of the square matrix A. inv (A)=A^ 1. Right Matrix division B/A=B*inv(A)=B*A^ 1 if B is a row vector and A is a square matrix. Left Matrix division A\B=inv(A)*B=A^ 1*B if B is a column vector and A is a square matrix. There are a lot of other build in functions for manipulating of matrices: See Gilat 3.5. ME 349, Engineering Analysis, Alexey Volkov 19

20 2.3. Advanced matrix manipulations in MATLAB Example 2.3.1: Calculate the matrix division for , The following six MATLAB expressions give the same result: B / A B * A^ 1 B * inv ( A ) ( A' \B')' ( (A')^ 1 * B' )' ( inv (A') * B' )' ans = Results can be different only due to round off error. ME 349, Engineering Analysis, Alexey Volkov 20

21 2.3. Advanced matrix manipulations in MATLAB Example 2.3.2: Calculate the matrix inverse for If : >> M * inv ( M ) ans = If : >> M * inv ( M ) ans = if 0then det 0 and the matrix is singular. if 0, but 0,thendet ~ 0, and calculation of the matrix inverse is very inaccurate due to round off errors. We must be extremely careful then calculating the matrix inverse for matrices with det 0! ME 349, Engineering Analysis, Alexey Volkov 21

22 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination System of linear equations (SLE) SLE with a square matrix of coefficients: When the solution exists and is unique Gauss elimination Reading assignment oflinear equations.html ME 349, Engineering Analysis, Alexey Volkov 22

23 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Systems of linear equations The system of linear equations (SLE) of equations with unknowns is the system : (2.4.1), coefficients of equations (1,..,, 1,..,)., unknowns (1,..,)., right hand sides (RHSs) of equations (1,..,). To solve a SLE means to find such that turn every equation in the system into identity. In general, a SLE may have/not have a solution. If a solution exits, it can be unique, or, alternatively, the system may have multiple solutions. ME 349, Engineering Analysis, Alexey Volkov 23

24 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Matrix form of a SLE (SLE in the matrix notation) Every SLE can be written in the matrix notation:,, (2.4.2), Matrix of coefficients of the SLE (of size )., Column vector of unknowns., Column vector of RHSs. The matrix form of the SLE given by Eq. (2.4.1): (2.4.3) Solution of an SLE with the square matrix of coefficients We will consider the most important case, when the number of equations is equal to the number of unknowns, and the matrix of coefficients is a square matrix. Theorem: If, then the unique solution of the SLE, Eq. (2.4.3) exists if det 0. Proof: If det 0, then the square matrix has the inverse, such that. ME 349, Engineering Analysis, Alexey Volkov 24

25 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Let s then multiple Eq. (2.4.3) from the left by : (2.4.4) Now we see that al least one solution given by Eq. (2.4.4) exists. In order to prove the uniqueness, let s assume that another solution exists, such that. But then. If we multiply the last equation by, then we obtain, i.e. and the solution is unique. Consequences: 1. Eq. (2.4.4) shows us how to find the solution of a SLE with the square matrix of coefficients: First we need to find the inverse of and then just to multiply it by the RHS vector. 2. A SLE given by Eq. (2.4.3) is called homogeneous if. The theorem says that the homogeneous SLE with det 0 has only the trivial solution. 3. If det 0 then the SLE can have multiple solutions or may not have them at all. ME 349, Engineering Analysis, Alexey Volkov 25

26 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Example: det 0 means that one or a few pairs of individual equations in the SLE are "proportional" to each other (this is not an accurate statement): The SLE has multiple solutions The SLE has no solutions In the majority of practical problems, however, the number of equations is very large, so it is enormously time consuming to find by applying Cramer's rule. The methods of solutions of SLEs focus on the problem: How to find the solution without explicit finding the inverse. There are two groups of methods, which have different domains of applicability: 1. Direct methods (Gauss elimination, Thomas algorithm for SLEs with tridiagonal matrices, etc). 2. Iterative methods (Jacobi method, Gauss Seidel method, etc). We will consider only Gauss elimination. ME 349, Engineering Analysis, Alexey Volkov 26

27 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Idea of a direct method: Step by step elimination of unknowns Question: How can we solve a SLE by hand? First equation: / Second equation: / / / / Can do this only if 0 / / / Third equation: Now we can substitute equations for and into the third equation and get equation with only one unknown. Once is determined, we can calculate and then. Gauss eliminations is the formalization of this approach in the form convenient for coding at arbitrary number of unknowns. ME 349, Engineering Analysis, Alexey Volkov 27

28 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Gauss elimination for a SLE with the square matrix of coefficients Let's first consider a SLE with the upper triangular matrix of coefficients (0if ): (2.4.5) ,, For the upper triangular matrix, det, so that the condition det 0 is equivalent to 0, i.e. all coefficients on the main diagonal are not equal to zero. Then the solution can be found with the back substitution procedure staring with as follows: /,, /,,, /,, etc until. (2.4.6) Conclusion: Once we have a SLE of form (2.4.5), its solution can be easily found without. ME 349, Engineering Analysis, Alexey Volkov 28

29 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination The Gauss elimination is the approach that reduces a SLE with the square matrix of coefficients and det 0 to an equivalent SLE with the upper triangular matrix in the form of Eq. (2.4.5). Two SLEs are called equivalent if they have the same solution. In order to transform a SLE to an equivalent SLE with the upper triangular matrix, we will use three rules: 1. Multiplying any equation of the SLE by a non zero constant does not change the solution. : Equivalent SLE if 0 2. Subtraction of one equation multiplied by a constant from another equation does not change the solution. : Equivalent SLE 3. Swapping any pair of equations (changing the order of equations) does not change the solution. : Equivalent SLE ME 349, Engineering Analysis, Alexey Volkov 29

30 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Now let's see how we can use these operation in order to transform the SLE of two equations to the SLE with the upper triangular matrix, where all elements of the main diagonal are equal to Check that If 0then swap equations. If both and are equal to zero, then det 0 and the Gauss elimination can not be applied. 3. Divide the first equation by : 4. Subtract the first equation multiplied by from the second equation:,, This is the equivalent SLE with the upper triangular matrix that can be easily solved by the back substitution. ME 349, Engineering Analysis, Alexey Volkov 30

31 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Now let's consider the general algorithm of the Gauss elimination applied to a SLE of equations. The Gauss elimination algorithm includes two sequential stages: 1. Forward sweep:reduction of the SLE to an equivalent SLE with the upper triangular matrix. 2. Back substitution given by Eqs. (2.4.6). Forward sweep of the Gauss elimination Consider all equations of the SLE equation by equation. For equation : 1. Check that current 0. If 0, find equation with 0() and swap equations and. 2. Divide equation by. 3. Subtract equation multiplied by from all subsequent equations with. 4. Go to the next equation 1. Equation 1 Equation ME 349, Engineering Analysis, Alexey Volkov 31

32 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination Implementation of the forward sweep in the MATLAB (simplified version without swapping equations) function [ Am, Bm ] = GaussElimForwardSweep ( A, B ) [ m n ] = size ( A ); Am = A; Bm = B; for i= 1 : n Bm(i) = Bm(i) / Am(i,i); Am(i,i+1:n) = Am(i,i+1:n) / Am(i,i); Am(i,i) = 1.0; for j = i+ 1 : n % Bm(j) = Bm(j) Bm(i) * Am(j,i); Am(j,i+1:n) = Am(j,i+1:n) Am(i,i+1:n) * Am(j,i); Am(j,i) = 0.0; end end end i= ME 349, Engineering Analysis, Alexey Volkov 32

33 2.4. Systems of linear equations (SLEs). Solution of a SLE by the Gauss elimination The Gauss elimination requires ~ arithmetic operations. Usually the Gauss elimination is used to solve SLEs with dense matrices of coefficients, i.e. the matrices where the majority of elements (entries) are not equal to zero. For SLEs with sparse matrices, where the majority of coefficients are 0, the Gauss elimination is computationally inefficient. For sparse matrices, iterative methods of solution of SLE are used, e.g. Gauss Seidel method, see. Seidel_method. Example: A sparse matrix of coefficients obtained when solving a finite element problem in two dimensions. The non zero elements are shown in black. The Gauss elimination can be used in order to find the matrix inverse. The basic property of the inverse matrix,,or is equivalent to individual equations with respect to unknown elements of matrix. The Gauss elimination for finding the inverse matrix is called the Gauss Jordan method. ME 349, Engineering Analysis, Alexey Volkov 33

34 2.5. Systems of linear equations with the tridiagonal matrix. The Thomas algorithm SLEs with band matrices of coefficients SLEs with the tridiagonal matrices of coefficients Thomas algorithm Implementation in the MATLAB Reading assignment ME 349, Engineering Analysis, Alexey Volkov 34

35 2.5. Systems of linear equations with the tridiagonal matrix. The Thomas algorithm In science and engineering, SLEs often have band matrix of coefficients. Example: Numerical solution of one dimensional heat conduction equation (we willconsider thisexample inpart3). A band (banded) matrix is a matrix whose non zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. The bandwidth of the matrix is the smallest number of adjacent diagonals to which the nonzero elements are confined. : Band matrix with the bandwidth equal to 4 It is very computationally inefficient to apply the general Gauss elimination method in order to solve a SLE with a band matrix of coefficients of size and bandwidth, since the absolute majority of arithmetic operations will be performed with zeros and will result in zero values. There are a lot of special algorithms of solving SLEs with band matrices that are derived from the Gauss elimination procedure and account for the special form of the band matrix of coefficients. ME 349, Engineering Analysis, Alexey Volkov 35

36 2.5. Systems of linear equations with the tridiagonal matrix. The Thomas algorithm A practically important case of a band matrix is the tridiagonal matrix, i.e. the band matrix with the bandwidth equal to 3, where all non zero elements are placed on the main diagonal and two additional diagonals above and below the main diagonal. The majority of elements in the tridiagonal matrix are equal to zero, so it is convenient to use new notation for coefficients and RHSs of a SLE with the tridiagonal matrix of coefficients: or where 0and 0 ME 349, Engineering Analysis, Alexey Volkov 36

37 2.5. Systems of linear equations with the tridiagonal matrix. The Thomas algorithm Now all non zero elements of the matrix of coefficients can be stored in three 1D arrays,,,,,,and,,. In order to solve the a SLE with the tridiagonal matrix of coefficients we also need the 1D array of RHSs,, Thomas Alg The Thomas algorithm (or tridiagonal matrix algorithm) is the modification of the Gauss elimination procedure for a SLE with the tridiagonal matrix. The Thomas algorithm consists of Forward sweep Back substitution ME 349, Engineering Analysis, Alexey Volkov 37

38 2.5. Systems of linear equations with the tridiagonal matrix. The Thomas algorithm Implementation of the Thomas algorithm in the MATLAB function [ x ] = ThomasAlg ( a, b, c, d ) [ m n ] = size ( b ) ; x = b; Forward sweep cc = c; dd = d; % Forward sweep cc(1) = c(1) / b(1); dd(1) = d(1) / b(1); for i= 2 : n coeff = b(i) a(i)* cc(i 1); cc(i) = c(i) / coeff; dd(i) = ( d(i) a(i)* dd(i 1) ) / coeff; end % Back substitution Back substitution x(n) = dd(n); for i= n 1 : 1 : 1 x(i) = dd(i) cc(i)* x(i+1); end end See another implementation of the Thomas algorithm at tridiag m/content/tridiag.m ME 349, Engineering Analysis, Alexey Volkov 38

39 2.6. Solution of SLE in the MATLAB. Problems Different approaches for solving SLE in the MATLAB Problem 1 Problem 2 Problem 3 Reading assignment Gilat, 3.3 ME 349, Engineering Analysis, Alexey Volkov 39

40 2.6. Solution of SLE in the MATLAB. Problems Consider a SLE where Different approaches for solving SLE in the MATLAB,, Assume we programmed as a 2D array A, as a column vector B We can solve the SLE in MATLAB 1. Via the inverse matrix 2. Via the matrix division X = inv ( A ) * B = A^ 1 * B X = A \B = ( B' / A' )' 3. Via the build in MATLAB function linsolve X = linsolve ( A, B ) ME 349, Engineering Analysis, Alexey Volkov 40

41 2.6. Solution of SLE in the MATLAB. Problems 4. Via the build in MATLAB function mldivide Notes: X = mldivide ( A, B ) mldivide is completely equivalent to the left matrix division: mldivide ( A, B ) = A \ B. mldivide analyzes the type of the matrix of coefficients and employs different solvers to handle different kinds of coefficient matrices. linsolve has an optional parameter opts: linsolve ( A, B, opts ). This parameter allows one to specify the type of the matrix ( upper triangular, etc. ) and to speed up calculations for a specific type of matrices. See mldivide and linsolve can be used for systems with non square matrices or for system with square matrix with zero determinants. In this case they return not actual, but some "generalized" solution. Anyway, its is a good practice to check that X is an actual solution, i.e., calculate A * X Band see that it is the zero vector. Multithreaded (parallelized) versions of mldivide and linsolve are available for the use on modern CPUs. mldivide and linsolve show significant increase in speed on large doubleprecision arrays (on order of 10,000 elements or more) when multithreading is enabled. ME 349, Engineering Analysis, Alexey Volkov 41

42 2.6. Solution of SLE in the MATLAB. Problems 5. Via multiple iterative methods Iterative methods are effective if the matrix of coefficients is large and sparse. Different methods are optimized for SLE with some particular matrices of coefficients Iterative methods are numerical methods, so they are able to find solution only with some finite error, which is usually controlled by the tolerance. The following iterative methods are available: pcg bicg bicgstab bicgstabl cgs gmres lsqr minres qmr symmlq tfqmr Preconditioned conjugate gradients method Biconjugate gradients method Biconjugate gradients stabilized method BiCGStab(l) Method Conjugate gradients squared method Generalized minimum residual method LSQR method Minimum residual method Quasi minimal residual method Symmetric LQ method Transpose free QMR method Iterative methods have a lot of adjustable parameters ( tolerance, etc ), but all these parameters can be set up by default: X = bicg ( A, B ) ME 349, Engineering Analysis, Alexey Volkov 42

43 2.7. Summary For the exam we must know how To make basic matrix operations (scalar multiplication, addition, matrix multiplication, and matrix transpose) by hand and in the MATLAB. To define special matrixes (column vector, row vector, square, zero, identity, symmetric, skew symmetric, upper and low triangular) in the MATLAB To calculated the determinants of matrices 2 x 2 and 3 x 3 by hand and determinants of arbitrary square matrices with the det MATLAB function. Tocalculatedtheinverseofmatrices2x2and3x3byhandwithCramer'sruleand inverse of arbitrary square matrices with the inv MATLAB function. Formulate and use major properties of determinants and inverse matrices. To write a system of linear equations (SLE) in the matrix notation. Formulate the idea of the Gauss elimination algorithm. To implement the back substitution step of the Gauss elimination in the MATLAB. To solve SLE in the MATLAB by utilizing left and right matrix division, calculation of the matrix inverse, and functions linsolve and mldivide. To implement the Tomas algorithm for solving the SLE with tridiagonal matrix of coefficients in the MATLAB ME 349, Engineering Analysis, Alexey Volkov 43