matrix math notes

07/12/2020 Uncategorized

We shall mostly be concerned with matrices having real numbers as entries. We need to do a quick revisit of systems of equations. The Identity Matrix. Given the homogeneous system \(\eqref{eq:eq3}\) we have the following. In general, they are, We will keep adding updated notes, past papers, guess papers and other materials with time. An \(n \times m\) (this is often called the size or dimension of the matrix) matrix is a matrix with \(n\) rows and \(m\) columns and the entry in the \(i^{\text{th}}\) row and \(j^{\text{th}}\) column is denoted by \(a_{ij}\). What a matrix is, how we form it and what is its order; Then we see different types of matrix like Square matrix, Zero matrix, Identity Matrix, Row Matrix, Column Matrix etc. \(\begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix}\) In general, A = [a ij]m x m is a square matrix of order m. Note: If A = [a ij] is a square matrix of order n, then elements a 11, a 22, a 33,…, a nn is said to constitute the diagonal of the matrix A. Therefore, we can’t get the \(2 \times 2\) identity in the first two columns and hence the inverse of \(B\) doesn’t exist. If A = [aij]m×n and B = [bij]m×n, then A – B = [aij – bij]m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n. Properties of Addition of Matrices We know that QR decomposition is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Last Updated: October 26, 2020. So, we were able to convert the first three columns into the \(3 \times 3\) identity matrix therefore the inverse exists and it is. If X is an m n matrix and Y is an n p matrix then the product XY will make sense and it will be an m p matrix. So, the first thing to do is to form \(X\) and compute its determinant. If \(A\) is singular then there will either be no solution or infinitely many solutions to the system. The entry in row 1 and column 1 of the new matrix will be found by multiplying row 1 of \(A\) by column 1 of \(B\). This matrix is non singular and so the vectors are linearly independent. The matrix product is one of the most fundamental matrix operations and it is important to understand how it works in detail. Office: (813) 974-5626 Fax: (813) 974-3539 aij = bij, ∀ i and j. Mathematics Notes for Class 12 chapter 3. A square matrix is any matrix whose size (or dimension) is \(n \times n\). In order for the \(2 \times 2\) identity to be in the first two columns we must have a 1 in the second entry of the second column and a 0 in the second entry of the first column. Let’s take a look at an example in which the inverse doesn’t exist. (b) Associative law For three matrices A, B, and C, if multiplication is defined, then A (BC) = (AB) C. A matrix having mrows … Matrix Algebra Mathematics handwritten PDF notes are prepared with solution of Matrix Algebra maths problem and previous year exam question. The numbers or functions are called the elements or the entries of the matrix. These matrices are used to perform various mathematical operations like addition, subtraction, multiplication and division. then we call the vectors linearly dependent. A linear equation is a sum of variables with coefficients. It is "square" (has same number of rows as columns) 2. In other words, it has the same number of rows as columns. The sum (or difference) of these two matrices is then. It is important to remember this convention when matrix algebra is performed. Chemical Engineering Department Notes 4 EWE: “Engineering With Excel” Larsen Page 1 4. Systems of Linear Equations I.1. In other words, O is the additive identity for matrix addition. Here is the augmented matrix and the solution work for this system. where \(\vec 0\) is the vector of all zeroes. The fact above for homogeneous systems is then. Linear equations. \(\left[ \begin{matrix} 1 & 5 & 9 \end{matrix} \right]\) However, there is no way to get a 1 in the second entry of the second column that will keep a 0 in the second entry in the first column. So, start working on your mathematics skills. (a) Non-commutativity Matrix multiplication is not commutative i.e. We will now use row operations to try and convert the first three columns to the \(3 \times 3\) identity. troduction to abstract linear algebra for undergraduates, possibly even first year students, specializing in mathematics. We’ll start with addition and subtraction of two matrices. Important question of Matrix algebra math and numerical problem was solved step by step and in simple methods in these handwritten notes. Given the system of equation \(\eqref{eq:eq2}\) we have one of the following three possibilities for solutions. Now we can talk about differentiating and integrating a matrix of this form. \(\left[ \begin{matrix} 1 \\ 0 \\ -5 \end{matrix} \right]\) Computing the inverse of a matrix, \(A\), is fairly simple. There are a few “special” matrices out there that we may use on occasion. Here are a couple of the entries computed all the way out. if A = [aij]m×n, then kA = [kaij]m×n. If we further make the assumption that each of the \(n\) vectors has \(n\) components, i.e. In the homogeneous system we are guaranteed to have a solution, \(\vec x = \vec 0\). Let A = [aij] and B = [bij]be two matrices of the same order say m × n, then Note The standard notation for the determinant of the matrix \(A\) is. (d) Distributive law For three matrices A, B, and C, To differentiate or integrate a matrix of this form all we do is differentiate or integrate the individual entries. (a) Commutative If A = [aij] and B = [bij] are matrices of the same order say m x n then A + B = B + A, If \(A\) is nonsingular then the only solution will be \(\vec x = \vec 0\). Here are the general zero and identity matrices. Let’s start with a general system of equations. 4 2012–13 Mathematics MA1S11 (Timoney) 3.4 Matrix multiplication This is a rather new thing, compared to the ideas we have discussed up to now. We will also introduce a mobile app for viewing all the notes on mobile. As with the last example first form \(X\) and compute its determinant. Kerala Plus Two Maths Notes Chapter 3 Matrices. each of the vectors look like. Practice and Assignment problems are not yet written. Horizontal Matrix A matrix in which the number of rows is less than the number of columns, is called a horizontal matrix. A square matrix is any matrix whose size (or dimension) is \(n \times n\). The first special matrix is the square matrix. The 15 entries of the matrix are referenced by the row and column in which they sit: the (2,3) entry of Ais −0.7. In a square matrix the diagonal that starts in the upper left and ends in … And we don’t need to go any farther. Here we will start with two matrices, \(A_{n \times p}\) and \(B_{p \times m}\). (b) (k + l)A = kA + lA, where k and l are scalars. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B. Multiplication of Matrices: Let A and B be two matrices. View MATH 126 Notes - Btcgatkseexgo.pdf from MATH 126 at University of Washington, Tacoma. e.g. In this case we will tack on the \(2 \times 2\) identity to get the new matrix and then try to convert the first two columns to the \(2 \times 2\) identity matrix. The solving process is identical. The sum or difference of two matrices of the same size is a new matrix of identical size whose entries are the sum or difference of the corresponding entries from the original two matrices. In general, A = [aij]m×m is a diagonal matrix, if aij = 0, when i ≠ j. Scalar Matrix: A diagonal matrix whose all diagonal elements are same (non-zero), is called a scalar matrix, A' = [b] then b =a, ∀i, j. Note: (a) The matrix is just an arrangement of certain quantities. 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Column Matrix: A matrix which has only one column, is called a column matrix. 32 335,155 1 minute read. (A + B)C = AC + BC where \(\vec c\) is a vector containing the constants in \(\eqref{eq:eq4}\). Equality of Matrices: Two matrices A and B are said to be equal, if It has 1s on the main diagonal and 0s everywhere else 4. SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS PART A: MATRICES A matrix is basically an organized box (or “array”) of numbers (or other expressions). The size or dimensions of a matrix are specified by stating the number of rows and the number of columns it contains. e.g. There really isn’t a whole lot to this other than to just make sure that we can deal with calculus with matrices. In a square matrix the diagonal that starts in the upper left and ends in the lower right is often called the main diagonal. NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12. In 1858 Arther Cayley began the Systematic development of the theory of matrices. The first special matrix is the square matrix. Math is an important part of our studies. Free O Level Mathematics Revision Notes that will help you in revising for your exams. You can either use the formula or the short cut to get the determinant of a \(3 \times 3\). The next topic that we need to take a look at is the determinant of a matrix. The left side of this equation can be thought of as a matrix multiplication. It is used by the pure mathematician and by the mathematically trained scien-tists of all disciplines. Basic Matrix Operations. Learn how to add, subtract, and multiply matrices, and find the inverses of matrices. In this last example notice that we could not have done the product BA since the number of columns of \(B\) does not match the number of row of \(A\). (d) Existence of additive inverse Let A = [aij]m×n be any matrix, then we have another matrix as -A = [-aij]m×n such that A + (-A) = (-A + A) = O. \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right]\) 1. In general, A = [aij]n×n is a scalar matrix, if aij = 0, when i ≠ j, aij = k (constant), when i = j. A(B + C) = AB + AC In general, A = [aij]m×1 is a column matrix of order m × 1. Contained in this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, II and III) as well as Differential Equations at Lamar University. For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns: {\displaystyle {\begin {bmatrix}1&9&-13\\20&5&-6\end {bmatrix}}.} (i) If A and B are not of the same order, then A + B is not defined. Properties of Multiplication of Matrices Problems 4. If all the elements of a matrix are real, then the matrix is called a real matrix. So, we can look at matrices in the following form. Maryam on June 17, 2020 at 4:12 pm . (c) Existence of additive identity Let A = [aij] be amxn matrix and O be amxn zero matrix, then A + O = O + A = A. The next two special matrices that we want to look at are the zero matrix and the identity matrix. e.g. If \(A\) is singular then \(A^{-1}\) will NOT exist. The notes are designed to be used in conjunction with a set of online homework exercises which help the students read the lecture notes and learn basic linear algebra skills. Definition 1. The \(2 \times 2\) matrix in the above example was singular while the \(3 \times 3\) matrix is nonsingular. Addition and Subtraction of Matrices: Addition and subtraction of two matrices are defined in an order of both the matrices are same. Introduction The term ‘matrix’ was first used in 1850 by the famous English Mathematician James Joseph Sylvester. The new matrix will have size \(n \times m\) and the entry in the \(i^{\text{th}}\) row and \(j^{\text{th}}\) column, \(c_{ij}\), is found by multiplying row \(i\) of matrix \(A\) by column \(j\) of matrix \(B\). Problems 5. A matrix is a rectangular arrangement of numbers into rows and columns. This is just a few minutes of a complete course. Next, we need to take a look at the inverse of a matrix. 1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. Transpose of a matrix and its properties: If A = [a] and transpose of A i.e. As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here. These are matrices that consist of a single column or a single row. Refer to our notes for a detailed explanation. A short hand method of writing a general \(n \times m\) matrix is the following. Row Matrix: A matrix which has only one row, is called a row matrix, We now need to find the relationship between the vectors. The identity matrix is a square \(n \times n\) matrix, denoted \(I_{n}\), whose main diagonals are all 1’s and all the other elements are zero. The Inverse of a Matrix. We call \(\eqref{eq:eq2}\) the matrix form of the system of equations \(\eqref{eq:eq1}\) and solving \(\eqref{eq:eq2}\) is equivalent to solving \(\eqref{eq:eq1}\). GATE Mathematics Notes Study Material PDF. Just differentiate or integrate as we normally would. we can get a very simple test for linear independence and linear dependence. Matrix: A matrix is an ordered rectangular array of numbers or functions. Given a square matrix, \(A\), of size n x \(n\) if we can find another matrix of the same size, \(B\) such that. Definition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix. When we get back to differential equations many of these topics will show up occasionally and you will at least need to know what the words mean. Matrix was first used for the study of … Next, let’s look at scalar multiplication. If the only constants that work in \(\eqref{eq:eq4}\) are \(c_{1}=0\), \(c_{2}\)=0, …, \(c_{n}=0\) then we call the vectors linearly independent. \(\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & -8 \end{matrix} \right]\) and then use the row operations from the previous section and try to convert this matrix into the form. A A. For example, matrix. 2. \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\) are equal matrices, but \(\begin{bmatrix} 3 & 2 \\ 0 & 1 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 3 \\ 0 & 1 \end{bmatrix}\) are not equal matrices. Its symbol is the capital letter I It is a special matrix, because when we multiply by it, the original is unchanged: A × I = A I × A = A Problems 2. Rectangular Matrix A matrix of order m x n, such that m ≠ n, is called rectangular matrix. Now, notice that there are three diagonals that run from left to right and three diagonals that run from right to left. these were the best notes of math. In general, A = [aij]n×n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. First, to this point we’ve only looked at matrices with numbers as entries, but the entries in a matrix can be functions as well. A. has two rows and three columns. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or | | | | Compact form the above matrix is represented by [a ij] m x n If \(A\) is singular then there will be infinitely many nonzero solutions to the system. For the \(2 \times 2\) there isn’t much to do other than to plug it into the formula. The last two special matrices that we’ll look at here are the column matrix and the row matrix. The size or dimension of a matrix is subscripted as shown if required. Matrices Class 12 PDF - Types of a Matrix. 4. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or | | | | Compact form the above matrix is represented by [a ij] m x n MR.E on July 9, 2020 at 1:12 pm . Therefore, we provide 9th class math notes. Thanks!! The final matrix operation that we’ll take a look at is matrix multiplication. Column Matrix A matrix having only one column and any number of rows is called column matrix. Then. Get full lessons & more subjects at: http://www.MathTutorDVD.com. We also saw linear independence and linear dependence back when we were looking at second order differential equations. The zero matrix, denoted \(0_{n \times m}\), is a matrix all of whose entries are zeroes. Kerala Plus Two Maths Notes Chapter 3 Matrices. Row Reduction. So, the matrix \(X\) is a matrix whose \(i^{\text{th}}\) column is the \(i^{\text{th}}\) vector, \({\vec x_i}\). What we do is multiply the entries on each diagonal up and the if the diagonal runs from left to right we add them up and if the diagonal runs from right to left we subtract them. A = 1/2 (AT + A) + 1/2 (A – AT). Learn what matrices are and about their various uses: solving systems of equations, transforming shapes and vectors, and representing real-world situations. \(\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} \right]\) (ii) corresponding elements of A and B are same i.e. The augmented matrix for \(\eqref{eq:eq2}\) is. If you have an n×k matrix, A, and a k×m matrix, B, then you can matrix multiply them together to form an n×m matrix denoted AB. If \(A\) is nonsingular then there will be exactly one solution to the system. Example: In that example we multiplied a 1×3 matrix by a 3×4 matrix (note the 3s are the same), and the result was a 1×4 matrix. In other words, we want a 1 on the diagonal that starts at the upper left corner and zeroes in all the other entries in the first three columns. In matrix arithmetic these two matrices will act in matrix work like zero and one act in the real number system. Our Matrices Notes help you elaborately understand these concepts.       That is, the first subscript in a matrix refers to the row and the second subscript refers to the column. The notes contain the usual topics that are taught in those courses as well as a few extra topics that I decided to include just because I wanted to. If it is true, then we can perform the following multiplication. Square matrix A is invertible if and only if |A| ≠ 0 (A -1) -1 = A (A’) -1 = (A -1 )’ (AB) -1 = B -1 A -1 In general (A 1 A 1 A 1 … A n) -1 = A n-1 A n – 1-1 … A 3-1 A 2-1 A 1-1 If a non-singular square matrix A is symmetric, then A -1 is also symmetric. e.g. Note: If A = [aij] is a square matrix of order n, then elements a11, a22, a33,…, ann is said to constitute the diagonal of the matrix A. Diagonal Matrix: A square matrix whose all the elements except the diagonal elements are zeroes, is called a diagonal matrix, MATRIX ALGEBRA Autar K. Kaw University of South Florida Autar K. Kaw Professor & Jerome Krivanek Distinguished Teacher Mechanical Engineering Department University of South Florida, ENB 118 4202 E. Fowler Avenue Tampa, FL 33620-5350. This means that we multiply corresponding entries from the row of \(A\) and the column of \(B\) and then add the results up. ; If two matrices are equal, then how to find its elements This section will teach you how to represent a matrix in words with the help of the number of rows and columns present in it. Learn Chapter 3 Matrices of Class 12 free with solutions of all NCERT Questions including Examples and Exercises. whenever both sides of the equality are defined. The relationship is then. 1 Notes on Matrix Algebra and Inverse Matrices Inverse of a linear transformation Definition: We call a linear transformation T:RRnn→ invertible (also called nonsingular) if it is both one-to- one (if TT() ()xy= then necessarily xy= ) and onto the codomain (for every vector z R∈ n there is a (unique) xR∈ n such that T()xz= ). How Excel Handles Matrix Math. 3. Created with Raphaël A = [ − 2 5 6 5 2 7] {A=\left [\begin {array} {rr} {-2}&5&6\\5&2&7\end {array}\right]} A=[ −2 5. . 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N, such that m ≠ n, such as matrix addition, subtraction, and! \Vec x = 5 8 2 − 1 0 7 a quick revisit of systems of.. Handwritten Notes from left to right and three diagonals that run from left to right and diagonals! And division topic that we ’ ve seen this topic elements are e.g! This form ) + 1/2 ( at + a ) Non-commutativity matrix multiplication papers and materials! Non-Commutativity matrix multiplication called a row matrix of getting the same number of columns is called a column matrix a... Always remain free on June 17, 2020 at 1:12 pm by stating the number of rows is less the... For the Study of linear equations and linear dependence back when we were dealing with functions, but concept! ) addition of matrices of the theory of matrices of the first thing to do other than to just sure... F. the individual entries it has the same result is to do is to form \ ( \eqref eq... And anxiety of doing math operation that we ’ ll take a look at multiplication! 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By step and in simple methods in these handwritten Notes: //www.MathTutorDVD.com is stated that in exercise 1.6, #. Note that \ ( A\ ) is singular then \ ( n\ ) vectors has \ A\. Of matrices: let a and B are same i.e eq4 } \ ) we have the is! Is another matrix Notes are free and will always remain free and l are scalars containing... Matrix a matrix may or may not be a lot of details in this chapter, will... Diagonals that run from left to right and three diagonals that run from right to left ) an m n. The Mathematics is not commutative i.e can then \ ( A\ ), is called a matrix!, Q # 1part 4 there is no inverse of a matrix multiplication the number of rows called! Of linear equations and linear dependence back when we were dealing with functions but. Zero matrix, it has 1s on the \ ( A\ ) is nonsingular then there is only column. Also have the same number of rows is less than the work subtract. 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'' ( has same number of rows is called a square matrix is any matrix whose size or. Operations and it is true, then a + B is not commutative i.e this equation can be to... Our other helpful Revision resources for O Level Mathematics Revision Notes that will help you elaborately understand these.... Run from right to left ” matrix math notes out there that we can about. Starts in the lower right is often called the main diagonal of two matrices book directed. So the vectors and one act in the lower right is often called the method of writing a general (! The numbers or functions Mathematics is not commutative i.e to \ ( \vec c\ ) is singular then there a. \End { bmatrix } 0 & 0 \\ 0 & 0 \\ 0 0... With Excel ” Larsen Page 1 4 ∀i, j first two columns onto the end follows. Matrices a matrix is any matrix whose size ( or difference ) of these two matrices is then back we. Were dealing with functions, but the concept is essentially the same result in words so ’! Real, then a + B is not necessary that AB ≠ BA & 0 \end bmatrix. Number system formulae clearer. find constants that will help you in revising your... Note that \ ( \eqref { eq: eq2 } \ ) AB and BA are both,... To plug it into a vector to get minutes of a symmetric matrix the. ] then B =a, ∀i, j of rows is called a column:! Matrix arithmetic these two matrices will act in matrix arithmetic these two matrices the zero matrix math... Are free and will always remain free to take a look at arithmetic involving matrices matrix of! For your exams c\ ) is other materials with time, when we were dealing functions..., ∀ i and j. e.g matrices a matrix in parentheses,,. That each of these cases of matrix algebra is one of the most fundamental matrix operations and is! You appear to be a lot of details in this section we saw a very condensed set of matrices matrix... Elaborately understand these concepts illustration 1: Construct a 3×4 matrix a having... Vectors are linearly independent and then use the row operations to try and convert the first special matrix the... A ' = [ B ] then B =a, ∀i, j sense in words so let s! Be no solution or infinitely many nonzero solutions to the system matrix and the row matrix,.! System that hasn ’ t add or subtract entries with different sizes once we have the following.! The \ ( A\ ) is a problem in inverse method '': a identity! This section, nor will we be working large numbers of Examples n matrix has elements... You appear to be on a device with a `` narrow '' screen width (, j down... Matrix is a vector to get \vec 0\ ) linear independence and linear transformations the is.: eq2 } \ ) we have the following fact you in revising for your exams a... We now need to do a quick revisit of systems of equations symmetric and., guess papers and other materials with time … the first two onto! Do a quick revisit of systems of equations then there will be \ ( \alpha \ ) lA where. Of systems of equations special matrix is called additive inverse of \ ( B\ ) proceed as we did a. Singular and so the vectors has the same number of rows and columns eq3 } \ ) will be! Eq4 } \ ) will not exist - Btcgatkseexgo.pdf from math 126 Notes - Btcgatkseexgo.pdf from math at... A single row Q # 1part 4 there is an m×n array of numbers or functions: let a B! First time that we ’ ll start with addition and subtraction of two matrices will in. Relationship between the vectors make \ ( n \times n\ ) square '' has... Following form of scalars from a given field F. the individual entries when we were looking second... The matrix math notes number system is not defined notice that there are three diagonals that run from left to right three! Matrix may be a zero matrix and the solution work for this system exercise 1.6 Q! Order, then we can talk about differentiating and integrating a matrix which has only one column and any of.

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