Matrix Math Rules Pdf

07 Aug, 2021

In this chapter we will typically assume that our matrices contain only numbers. MATRIX ALGEBRA Scalar Multiplication of Matrices.

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The diagonal matrix diag111 is called the identity matrix and is usually denoted by I n 10.

Matrix math rules pdf. Mathguyus Developed specifically for math students from Middle School to College based on the authors extensive experience in professional mathematics in a business setting and in math tutoring. If X is a matrix then we can say X 0 X if 0 means the zero matrix of the same size as X. That is for R aB then r ij ab ij for all i and j.

Behaviour of nature at its core. What a matrix mostly does is to multiply a vector x. A basic result of linear algebra is that AA1 I.

But A 1 might not exist. Here each element in the product matrix is simply the scalar multiplied by the element in the matrix. Their product is the identity matrixwhich does nothing to a vector so A 1Ax D x.

A unit matrix is one with all elements zero except the leading diagonal in which the elements are 1. Multiplying Ax D b by A 1 gives A 1Ax D A b. If you have an nk matrix A and a km matrix B then you can matrix multiply them together to form an nm matrix denoted AB.

Deﬁnition The transpose of an m x n matrix A is the n x m matrix AT obtained by interchanging rows and columns of A Deﬁnition A square matrix A is symmetric if AT A. DetA a 11a 22 a nn. There are several rules for matrix multiplication.

Some of these are1 ABBA cABcAcB ABCABC CAB CACB ABCACBC ABC ABC Perhaps the most interesting and unexpected of the above rules is ABG ABC. The rules can be stated in terms of elementary matrices as follows. Whatever A does A 1 undoes.

This has a special name. 0 diag00 is called the zero matrix. A little algebra you can verify the following manipulation rules.

The product of the matrices Aa ij and Bb jk of orders mnand nprespectively is the matrix AB Cc ik of order mpwhose generic element is c ik P j a ijb jka i1b 1ka i2b. The product A A is like multiplying by. If we wanted to make the notation less.

Matrix transpose AT 15 33 52 21 A 1352 532 1 Example Transpose operation can be viewed as ﬂipping entries about the diagonal. The material in this chapter will be covered in your Linear Algebra class Math 254 at Mesa. If a matrix is multiplied by a constant all the elements are multiplied by a constant.

1a A matrix A is said to be diagonal if a ij 0 i Wj. The product of the matrix Aa ij with an arbitrary scalar or number is the matrix Aa ij. We call this associativity and that matrix multiplication.

RAB rArB Scalar multiplication distributes over matrix addition 2. Suppose A is a square matrix. We see that if we multiply by k 0 we get a matrix where all the entries are 0.

MATRICES A matrix is basically an organized box or array of numbers or other expressions. The m n matrix where every entry is 0 is called the m n zero matrix. Matrix arithmetic has some of the same properties as real number arithmetic.

The determinant of the unit matrix is 1. We look for an inverse matrix A 1 of the same size such that A 1 times A equals I. AB C ABC Matrix addition is associative 3.

1022 sup Supremum of a set jjAjj Matrix norm subscript if any denotes what norm AT Transposed matrix A TThe inverse of the transposed and vice versa A T A 1T A. Thus we have zero matrices of every possible size. But since mathematics is the language of nature its required to quantify the prediction of quantum mechanics.

This present document has been put together to ease you into the mathematics of quantum mechanics. If an invertible matrix A has been reduced to rref form then its determinant can be found by det 1 1 2 s A k k k r where s is the number of row swaps performed and k1 k2 kr are the scalars by which rows have been divided. DiagA ij ijA ij eigA Eigenvalues of the matrix A vecA The vector-version of the matrix A see Sec.

Properties of Matrix Arithmetic Let A B and C be mn matrices and rs R. O tX 0 X tO sX tY sX tY X tYs X tY Y tX X tY X tY X tY X t Y tZ X tZ Y tZ distributive laws X tY Z X tY X tZ. RANK The number of leading 1s is the rank of the matrix.

It follows that the product of a constant A with the unit matrix is as shown. AB B A Matrix addition is commutative 2. Ie AT ij A ji ij.

The first concerns the multiplication between a matrix and a scalar. Swap If Eis an elementary matrix for a swap rule then detEA 1detA. MATRICES and SYSTEMS OF EQUATIONS PART A.

2 Laws of Matrix Arithmetic Many of the standard rules from ordinary arithmetic carry over into matrix arithmetic. Well use special mathematics complex numbers and linear algebra vectors and matrices. If A is square and square matrix F satisﬁes FA I then F is called the inverse of A and is denoted A1 the matrix A is called invertible or nonsingular if A doesnt have an inverse its called singular or noninvertible by deﬁnition A1A I.

This is a one-arrow Sarrus rule valid for dimension n. Triangular The value of detA for either an upper triangular or a lower triangular matrix Ais the product of the diagonal elements. We sometimes use AB for the matrix product if that helps to make formulae clearer The matrix product is one of the most fundamental matrix operations and it is important to understand how it works in detail.

In this book a matrix is denoted by an uppercase letter. Matrices An mn matrix is a rectangular array of mn scalars in m rows and n columns. 1b A diagonal matrix A may be denoted by diagd1d2d n where a ii d i a ij 0 j Wi.

TrA Trace of the matrix A diagA Diagonal matrix of the matrix A ie. Thus 8 2 6 3 7 16 48 24 56 Matrix multiplication involving a scalar is commutative. 2x3y7z 29 x1 7 2 x2 x3 x4 17x5 3 10 y1 14 2y 4 4 y2 13y1 7r.

This is x D A 1b. 01 or simply Iwhenn is assumed to be known.

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