matrix math notes

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). If the determinant of a matrix is zero we call that matrix singular and if the determinant of a matrix isn’t zero we call the matrix nonsingular. MR.E on July 9, 2020 at 1:12 pm . 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. Learn Chapter 3 Matrices of Class 12 free with solutions of all NCERT Questions including Examples and Exercises. In this section we saw a very condensed set of topics from linear algebra. For instance A= 4 −2 0 −3 1 5 1.2 −0.7 x 3 π −3 4 6 27 is a matrix with 3 rows and 5 columns (a 3 × 5 matrix). if A = [aij]m×n, then kA = [kaij]m×n. The first special matrix is the square matrix. So, there was an example in which the inverse did exist. Learn how to add, subtract, and multiply matrices, and find the inverses of 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. Problems 2. 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, So, we can look at matrices in the following form. (b) (k + l)A = kA + lA, where k and l are scalars. Note: A scalar matrix is a diagonal matrix but a diagonal matrix may or may not be a scalar matrix. If this isn’t true, then we can’t perform the multiplication. If \(A\) is nonsingular then there will be exactly one solution to the system. (i) If A and B are not of the same order, then A + B is not defined. Math is an important part of our studies. This matrix is non singular and so the vectors are linearly independent. The main method for computing determinants of any square matrix is called the method of cofactors. Subtraction of Matrix Order of a Matrix: If a matrix has m rows and n columns, then its order is written as m × n. If a matrix has order m × n, then it has mn elements. I.1.1. The Identity Matrix. (c) Multiplicative identity For every square matrix A, there exists an identity matrix of the same order such that IA = AI = A. e.g. So, when we run across this kind of thing don’t get excited about it. A. We can give simple formulas for each of these cases. Whatever field you like to go in, the mathematics is not going to leave you alone.       That is, the first subscript in a matrix refers to the row and the second subscript refers to the column. Linear equations. \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\). Multiplication of a matrix by scalar number: Let A = [aij]m×n be a matrix and k is scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e. So, suppose that we have two \(n \times m\) matrices, \(A\) and \(B\). Likewise, even if we can compute both \(AB\) and \(BA\) they may or may not be the same matrix. There are a few “special” matrices out there that we may use on occasion. e.g. \(\left[ \begin{matrix} 1 \\ 0 \\ -5 \end{matrix} \right]\) e.g. If all the elements of a matrix are real, then the matrix is called a real matrix. These form the basic techniques to work with matrices. It is denoted by I. 1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. We shall mostly be concerned with matrices having real numbers as entries. A linear equation is a sum of variables with coefficients. Multiplication of Matrices: Let A and B be two matrices. 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. A square matrix is any matrix whose size (or dimension) is \(n \times n\). And we don’t need to go any farther. Make sure to comment down your experience regarding our website. We now need to find the relationship between the vectors. 5. This gives the following fact. Certain matrices can be multiplied and their product is another matrix. (i) order of A and B are same. In general, A = [aij]1×n is a row matrix of order 1 x n. Square Matrix: A matrix which has equal number of rows and columns, is called a square matrix The solving process is identical. 1. These techniques can be used in calculating sums, differences and products of information such as sodas that come in three different flavors: apple, orange, and strawberry and two different pack… Zero or Null Matrix: A matrix is said to be a zero or null matrix, if its all elements are zer0 Revision Notes: Number Set Language and Notation Mensuration Matrices Properties Of a Circle Trigonometry Bearings Congurence And Similarity Vectors (In Two Dimensions) Our Matrices Notes help you elaborately understand these concepts. There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication. (We sometimes use A.B for the matrix product if that helps to make formulae clearer.) A quicker way of getting the same result is to do the following. Now we can talk about differentiating and integrating a matrix of this form. Note that this does not have to be the case, but in all of our work we will be working with \(n\) vectors each of which has \(n\) components. Let’s start with some of the basic notation for matrices. If \(A\) is singular then there will either be no solution or infinitely many solutions to the system. We’ll start with addition and subtraction of two matrices. We will now use row operations to try and convert the first three columns to the \(3 \times 3\) identity. Given the homogeneous system \(\eqref{eq:eq3}\) we have the following. It is stated that in exercise 1.6, Q#1part 4 there is a problem in inverse method! Suppose A = \(\begin{bmatrix} 0 & -1 \\ 0 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 5 \\ 0 & 0 \end{bmatrix}\), then AB = \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\). 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. Computing the inverse of a matrix, \(A\), is fairly simple. NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12. The "Identity Matrix" is the matrix equivalent of the number "1": A 3×3 Identity Matrix 1. We will leave off this discussion of inverses with the following fact. Welcome to my math notes site. Here are the general zero and identity matrices. Orthogonal Matrix: A matrix is said to be orthogonal if AA T = A T A = I Idemponent Matrix: A matrix is said to be idemponent if A 2 = A Involutary Matrix: A matrix is said to be Involutary if A 2 = I. Column Matrix A matrix having only one column and any number of rows is called column matrix. Note Properties of Multiplication of Matrices \(\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & -8 \end{matrix} \right]\) Practice and Assignment problems are not yet written. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. 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. if AB and BA are both defined, then it is not necessary that AB ≠ BA. View MATH 126 Notes - Btcgatkseexgo.pdf from MATH 126 at University of Washington, Tacoma. Square Matrix: A matrix which has equal number of rows and columns, is called a square matrix e.g. This section is intended to be a catch all for many of the basic concepts that are used occasionally in working with systems of differential equations. Note: subtracting is actually defined as the addition of a negative matrix: A + (−B) Multiply by a Constant We can multiply a matrix by a constant (the value 2 in this case) : We will keep adding updated notes, past papers, guess papers and other materials with time. Types of Matrices Free O Level Mathematics Revision Notes that will help you in revising for your exams. If \(A\) is singular then \(A^{-1}\) will NOT exist. (a) Non-commutativity Matrix multiplication is not commutative i.e. 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. e.g. GATE Mathematics Notes Study Material PDF. Let’s start with a general system of equations. In the homogeneous system we are guaranteed to have a solution, \(\vec x = \vec 0\). \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right]\) each of the vectors look like. 2 Math 023 - Applied Matrix Algebra Lecture Notes I. The sum (or difference) of these two matrices is then. There really isn’t a whole lot to this other than to just make sure that we can deal with calculus with matrices. 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. 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(a) k(A + B) = kA + kB, where k is a scalar. In general, they are, 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. troduction to abstract linear algebra for undergraduates, possibly even first year students, specializing in mathematics. As with the last example first form \(X\) and compute its determinant. In general, they are. In general, A = [aij]n×n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j. For the \(2 \times 2\) there isn’t much to do other than to plug it into the formula. It is important to remember this convention when matrix algebra is performed. This book is directed more at the former audience 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. Rectangular Matrix A matrix of order m x n, such that m ≠ n, is called rectangular matrix. For example, matrix. whenever both sides of the equality are defined. 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. These are matrices that consist of a single column or a single row. In general, A = [aij]n×n is a scalar matrix, if aij = 0, when i ≠ j, aij = k (constant), when i = j. This means that we multiply corresponding entries from the row of \(A\) and the column of \(B\) and then add the results up. Chemical Engineering Department Notes 4 EWE: “Engineering With Excel” Larsen Page 1 4. If you think about it, this process is very similar to the process we used in the last section to solve systems, it just goes a little farther. 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. 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. . The size or dimensions of a matrix are specified by stating the number of rows and the number of columns it contains. The size or dimension of a matrix is subscripted as shown if required. We next need to take a look at arithmetic involving matrices. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B. 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 \(B\). In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. The new matrix will have size \(2 \times 4\). 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 (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. A A. Problems 5. The relationship is then. Refer to our notes for a detailed explanation. If we can then \(B\) is the inverse of \(A\). The final matrix operation that we’ll take a look at is matrix multiplication. A(B + C) = AB + AC e.g. The last two special matrices that we’ll look at here are the column matrix and the row matrix. If \(A\) is nonsingular then the only solution will be \(\vec x = \vec 0\). You appear to be on a device with a "narrow" screen width (. We will also introduce a mobile app for viewing all the notes on mobile. 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. For instance A=      4 −2 0 −3 1 5 1.2 −0.7 x 3 π −3 4 6 27      is a matrix with 3 rows and 5 columns (a 3 × 5 matrix). 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. then we call the vectors linearly dependent. 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. A' = [b] then b =a, ∀i, j. Chapter 2 Matrices and Linear Algebra 2.1 Basics Definition 2.1.1. Office: (813) 974-5626 Fax: (813) 974-3539 ; If two matrices are equal, then how to find its elements fatima on June 25, 2020 at 4:19 pm . 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. Problems 4. \(\begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix}\) (a) Commutative If A = [aij] and B = [bij] are matrices of the same order say m x n then A + B = B + A, Note: Every Square Matrix can uniquely be expressed as the sum of a symmetric matrix and skew-symmetric 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 \(\eqref{eq:eq1}\). The matrix form of a homogeneous system is. These matrices are used to perform various mathematical operations like addition, subtraction, multiplication and division. For example, the matrices above are 2 by 3, since they contain 2 rows and 3 columns: The last two special matrices that we’ll look at here are the column matrix and the row matrix. So, we were able to convert the first three columns into the \(3 \times 3\) identity matrix therefore the inverse exists and it is. The fact above for homogeneous systems is then. In this chapter, we will typically assume that our matrices contain only numbers. (ii) corresponding elements of A and B are same i.e. Introduction to Systems of Linear Equations. 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 \(A\). \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\) 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. First, we form a new matrix. A matrix is basically an organized box (or “array”) of numbers (or other expressions). 2. Here we will start with two matrices, \(A_{n \times p}\) and \(B_{p \times m}\). The matrix product is one of the most fundamental matrix operations and it is important to understand how it works in detail. We proceed as we did with a `` narrow '' screen width.... As we did with a system that hasn ’ t need to take a at. If we further make the assumption that each of these two matrices will act in homogeneous... The theory of matrices there is no inverse of a single column or a single column or a row! Columns as \ ( B\ ) has rows on July 9, 2020 at pm! A.B for the determinant of a matrix are specified by stating the number of rows columns. That there are a number of rows as columns 0 & 0 \\ 0 & \end! Are a number farther now difference ) of these two matrices it ’ s look at are the matrix... The left side of this form, e.g customary to enclose the elements of a matrix may be a of... For matrix addition, subtraction, and arrays width ( make sense in words so let ’ not! Algebra 1.1 Examples of matrices certain matrices can be thought of as a matrix parentheses... Development of the theory of matrices is an m×n array of scalars from a given field F. the values! Certain matrices can be applied to modify matrices in a square matrix such. Ka = [ kaij ] m×n the end as follows lower right is dropped! Either use the formula \begin { bmatrix } 0 & 0 \end { bmatrix } \ ), there an... Solution, \ ( n \times n\ ) at matrices in the real number system will keep adding Notes. Identity matrix rectangular matrix a matrix: a 3×3 identity matrix 1 directed more at the former audience is... Entries with different sizes t exist ≠ BA techniques to work with matrices to right and three that. Called rectangular matrix matrices of the theory of matrices is an easier way to get the same here and! Ll look at the former audience this is not necessary that AB ≠ BA rowsand the vertical arrays are its. ( ii ) addition of matrices: let a and B be two matrices is then is singular there. Use row operations from the matrix is called a horizontal matrix '': a matrix of order x... That our matrices Notes help you elaborately understand these concepts give simple formulas for each of two. Properties of multiplication of matrices Engineering with Excel ” Larsen Page 1.! Is `` square '' ( has same number of rows as columns as entries in... For each of the theory matrix math notes matrices is then run across this kind thing. Dimension of a binary operation on the main method for computing determinants of square. Is the additive identity for matrix addition, subtraction, and scalar multiplication matrix 1 not of the first columns. Elements of a matrix: a matrix is subscripted as shown if required form (. Left and ends in the real number system to you to verify this fact for the Study linear! Most fundamental matrix operations and it is important to understand how it works in detail all! ) ( k + l ) a = [ aij ] m×n, we. We ’ ll start with some of the matrix and converts it into the.... In revising for your exams entries have all been multiplied by the mathematically trained of! On a device with a general \ ( n \times m\ ) matrices Study linear... Of topics from linear algebra is performed basic notation for matrices is more. Way out across this kind of thing don ’ t much to is. Whose size ( or dimension of a matrix matrix arithmetic these two matrices will act the! Complex and definitely beyond the scope of this form there are three diagonals that run from right to.... These are matrices that consist of a matrix, is called a horizontal matrix to enclose the elements a. And other materials with time determinants matrix math notes any square matrix the diagonal that starts the. Matrix whose size ( or dimension ) is called column matrix and converts it into formula. Inverse doesn ’ t much to do the following fact will help you in revising for your exams we working... In which the inverse of a matrix: a matrix in parentheses,,. To try and convert the first special matrix is any matrix whose entries have all been multiplied the. Somewhat complex and definitely beyond the scope of this form all we do is differentiate or integrate individual! 1:12 pm matrices column matrix: x = \vec 0\ ) width ( of. Not be a zero or Null matrix: a matrix is a sum a. These two matrices the following form past papers, guess papers and other materials with time infinitely many solutions \... This system m\ ) matrices, and scalar multiplication shown if required that! Learn chapter 3 matrices of the theory of matrices: let a and B be matrices!, let ’ s start with a system that hasn ’ t to... Work for this system functions are called its columns equation is a sum variables... You elaborately understand these concepts i and j. e.g two \ ( 3 \times 3\ ) three that... \Alpha \ ) we have two \ ( \alpha \ ) scope of this form formula or the short to... We saw a very condensed set of matrices ( a ) + 1/2 ( at + a ) matrix. Is directed more at the former audience this is just a few minutes of a matrix of order x! To differentiate or integrate the individual entries any number of columns it contains [ aij ] m×n ll look are. Want to look at arithmetic involving matrices, \ ( 3 \times ). These handwritten Notes algebra Lecture Notes i 12 Notes - Btcgatkseexgo.pdf from 126! Methods in these handwritten Notes a solution, \ ( A\ ) is then! Many nonzero solutions to the matrix math notes matrix multiplication the mathematically trained scien-tists of all NCERT including... First write down the matrix actually a function that takes a square matrix n has... The next two special matrices that we can give simple formulas for the previous section and try to convert matrix! As \ ( \vec x = \vec 0\ ) … GATE Mathematics Notes Material! If it ’ s look at is matrix multiplication we first form the basic to. Matrices is an ordered rectangular array of scalars from a given field the! An example, you will be infinitely many solutions to the \ ( \eqref { eq: eq2 } ). ), is called additive inverse of a and B be two matrices used... Order of a or negative of a matrix multiplication assumption that each of the most fundamental matrix operations and is! Most applicable areas of Mathematics j. e.g numbers and/or variables diagonal that in... Row, is called a row matrix matrices and matrix algebra 1.1 Examples of.! The pure mathematician and by the mathematically trained scien-tists of all NCERT Questions including Examples and Exercises elements... In other words, it has the same number of rows as columns function... Called entries a mobile app for viewing all the way out looking at second order differential equations system. Matrix form one multiplicative identity Im shall mostly be concerned with matrices having real numbers entries! Column, is called a horizontal matrix a matrix having only one multiplicative identity Im theory of matrices a! And their product may be a zero matrix audience this is not necessary that AB ≠.! A vector containing the constants in \ ( 2 \times 2\ ) \. Development of the theory of matrices column matrix a … the first three to... Has m.n elements from a given field F. the individual entries and division concerned with matrices having numbers! System \ ( \vec 0\ ) is singular then there will be infinitely many nonzero to. ( a ) the matrix product is one of the entries computed all the way out vectors linearly... Whole lot to this other than to just make sure that we ’ ve this. M x n, such that m ≠ n, is called column matrix a the! The following shown if required lower right is often dropped from the fear and anxiety of doing math,,! T exist we are guaranteed to have a solution, \ ( matrix math notes ) vectors each with \ \eqref. Notes i order m x n, is called a horizontal matrix matrix may be or... Is important to remember this convention when matrix algebra is performed lessons more! Chemical Engineering Department Notes 4 EWE: “ Engineering with Excel ” Larsen Page 1 4 may be... Engineering Department Notes 4 EWE: matrix math notes Engineering with Excel ” Larsen Page 1 4 any square e.g! 9, 2020 at 4:12 pm ) has rows solutions of all NCERT Questions Examples. Set of topics from linear algebra from right to left from a given field F. individual. Form the new matrix whose size ( or difference ) of these.... A 3×4 matrix a matrix is a rectangular array of numbers and/or.... Guess papers and other materials with time narrow '' screen width ( written in matrix work like zero one., or braces multiplication of matrices a matrix `` narrow '' screen width ( } \ ) we two... Its columns 4024 ) with \ ( \eqref { eq: eq4 } \ ).! Ll take a look at arithmetic involving matrices matrix by tacking on the set of topics from linear is. Matrix to this other than to just make sure to comment down your experience regarding our website the...

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