Properties Of Matrix Multiplication Pdf

04 Mar, 2021 Post a Comment

Matrix multiplication is associative that is ABC ABC. Furthermore if the vector norm is a p-norm then the induced matrix norm satis es the submultiplicative property.

Matrix Multiplication And Word Problems Worksheet 3

The matrix I is called the identity matrix and must have the same order as A.

Properties of matrix multiplication pdf. There are many ways of looking at matrix multiplication and. Matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second. Equally the matrix A is the inverse of the matrix B.

2 Matrix Multiplication The product of two matrices A R m n and B R n p is the matrix C AB R m p where C ij n summationdisplay k 1 A ik B kj. With 0 denoting the zero matrix 0A A0 0. For example matrix A is a 2 3 matrix and matrix B is a 3 4 matrix then AB is a 2 4 matrices.

AB B A commutative property 2. Properties of matrix operations The operations are as follows. Let A B C be matrices and let c be a scalar.

That is the 1-norm of a matrix is its maximum column sum. B CA BACA. Matrix Multiplication.

132 Multiplication Properties Provided that each of the following matrix products exists we have AB C AB AC B CA BACA ABC ABC However usually AB BA For a square matrix A we have. Thus trMN trNM for any square matrices Mand N. Suppose that A is an m n matrix and that in each of the following identities the sizes of B and C are compatible when necessary for the product to be defined.

AB C AB AC. Solution Using the rules of matrix multiplication AB 4 3 2 5 6 3 3 5 2 3 4 3 1 2 2 7 11 9 1 0 0 0 1 0 0 0 1 I. In this case the rref of A is the identity matrix denoted In characterized by the diagonal row of 1s surrounded by zeros in a square matrix.

AB C A BC 4. Properties of Scalar Multiplication. 6 NM 1 1 1 2.

Commutative Property. ABC ABCAssociativity of matrix mul-tiplication 5. This will allow me to prove some useful properties of these operations.

If A and B are matrices of the same size m n then A B their sum is a matrix of size m n. ABC ABACDistributivity of matrix multiplication 4. Rref A 1 0 0 0 1 0 0 0 1 LINEAR TRANSFORMATION.

AOmn A additive identity property 4. There are important properties which hold for real numbers but not for matrices. αβA αβA αABαAαB.

Outer parameters become parameters of matrix AB What sizes of matrices can be multiplied together. If A is a square matrix and k is a positive integer we deﬁne Ak A AA k factors Properties of matrix multiplication. Then the following properties hold.

In matrix multiplication the product of m n matrix and na matrix is the m a matrix. The following matrix norms are of particular interest. We can multiply a number aka.

2 1 6 9 3 6 0 2 12 18 6 12 0 sometimes you see scalar multiplication with the scalar on the right α βA αAβA. Suppose are matrices and are scalars. Properties of Matrix Arithmetic.

Lets look at them in detail We used these matrices. ABC AB C associative property 3. Theorem 3 Algebraic Properties of Matrix Multiplication 1.

When a vector is multiplied by an identity matrix of the same dimension the product is the vector itself Inv v. Scalar by a matrix by multiplying every entry of the matrix by the scalar this is denoted by juxtaposition or with the scalar on the left. KAk 1 max kxk 11 kAxk 1 max 1 j n Xm i1 ja ijj.

Now Ill give precise definitions of the various matrix operations. TrMN trX l Mi l N l j X i X l Mi l N l i X l X i Nl iM i l trX i Nl iM i l trNM. KA B kA kB Distributivity of scalar multiplication II 3.

For a square matrix A AI IA A where I is the identity matrix of the same order as A. A B C AB AC A B C AC BC 5. AO O OA and AI A IA where I is a diagonal matrix with ones on the main diagonal and zeros elsewhere.

While matrix multiplication does not commute the trace of a product of matrices does not depend on the order of multiplication. Properties of matrix multiplication. Ive given examples which illustrate how you can do arithmetic with matrices.

MN 2 1 1 1. In the previous example M 1 1 0 1N 1 0 1 1. U 32U0162O BKdu WtXae MSodfNtBwuafrKeE MLRLXCQH O QAjl PlF 1r siUg8h2t 4su crPeps9eHr0vOeld4.

If A is a matrix of size m n and c is a scalar then cA is a matrix of size m n. Note that in order for the matrix product to exist the number of columns in A must equal the number of rows in B. Multiplicative property of Zero.

K A kA A Distributivity of scalar multiplication I 2. BA 3 4 3 1 2 2 7 11 9 4 3 2 5 6 3 3 5 2 1 0 0 0 1 0 0 0 1 I. Matrix multiplication For m x n matrix A and n x p matrix B the matrix product AB is an m x p matrix.

The matrix B is the inverse of the matrix A and this is usually written as A1. 1 Properties of Addition and Scalar Multiplica-tion Theorem 1 Let AB and C be m n matrices let Omn denote the m n matrix whose entries are all zeros and let c and d be scalars real numbers. A B B ACommutativity of matrix ad-dition 6.

It is called the natural or induced matrix norm. If a matrix is multiplied by a zero matrix the result matrix is a zero matrix. If you look at the definitions youll see the ideas we showed earlier by example.

Q v xMPad8eB Bwqi lt Ih n yIRnzf Ui3n WiSt teD VAdl9gxe Gbnr saX S2MK Worksheet by Kuta Software LLC. Zero matrix on multiplication If AB O then A O B O is possible 3. If A is a matrix of size m n and B is a matrix of.

The following properties hold.

Intro To Matrix Multiplication Video Khan Academy