Properties Of Matrix Multiplication Pdf

21 Apr, 2021

This will allow me to prove some useful properties of these operations. Now Ill give precise definitions of the various matrix operations.

Algebra 2 Worksheets Matrices Worksheets

Ive given examples which illustrate how you can do arithmetic with matrices.

Properties of matrix multiplication pdf. K A kA A Distributivity of scalar multiplication I 2. AB B A commutative property 2. 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.

If A and B are matrices of the same size m n then A B their sum is a matrix of size m n. There are many ways of looking at matrix multiplication and. A B C AB AC A B C AC BC 5.

Lets look at them in detail We used these matrices. Matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second. AOmn A additive identity property 4.

ABC ABCAssociativity of matrix mul-tiplication 5. That is the 1-norm of a matrix is its maximum column sum. 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.

Matrix multiplication is associative that is ABC ABC. Multiplicative property of Zero. Outer parameters become parameters of matrix AB What sizes of matrices can be multiplied together.

For example matrix A is a 2 3 matrix and matrix B is a 3 4 matrix then AB is a 2 4 matrices. Q v xMPad8eB Bwqi lt Ih n yIRnzf Ui3n WiSt teD VAdl9gxe Gbnr saX S2MK Worksheet by Kuta Software LLC. The following properties hold.

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. KA B kA kB Distributivity of scalar multiplication II 3. Zero matrix on multiplication If AB O then A O B O is possible 3.

In matrix multiplication the product of m n matrix and na matrix is the m a matrix. U 32U0162O BKdu WtXae MSodfNtBwuafrKeE MLRLXCQH O QAjl PlF 1r siUg8h2t 4su crPeps9eHr0vOeld4. It is called the natural or induced matrix norm.

AO O OA and AI A IA where I is a diagonal matrix with ones on the main diagonal and zeros elsewhere. A B B ACommutativity of matrix ad-dition 6. With 0 denoting the zero matrix 0A A0 0.

If A is a square matrix and k is a positive integer we deﬁne Ak A AA k factors Properties of matrix multiplication. 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. Matrix Multiplication.

When a vector is multiplied by an identity matrix of the same dimension the product is the vector itself Inv v. Rref A 1 0 0 0 1 0 0 0 1 LINEAR TRANSFORMATION. If a matrix is multiplied by a zero matrix the result matrix is a zero matrix.

Thus trMN trNM for any square matrices Mand N. In the previous example M 1 1 0 1N 1 0 1 1. B CA BACA.

We can multiply a number aka. ABC AB C associative property 3. MN 2 1 1 1.

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. Then the following properties hold. There are important properties which hold for real numbers but not for matrices.

Matrix multiplication For m x n matrix A and n x p matrix B the matrix product AB is an m x p matrix. Commutative Property. ABC ABACDistributivity of matrix multiplication 4.

If you look at the definitions youll see the ideas we showed earlier by example. Note that in order for the matrix product to exist the number of columns in A must equal the number of rows in B. Properties of Matrix Arithmetic.

AB C AB AC. 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. 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.

Furthermore if the vector norm is a p-norm then the induced matrix norm satis es the submultiplicative property. 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. If A is a matrix of size m n and B is a matrix of.

αβA αβA αABαAαB. If A is a matrix of size m n and c is a scalar then cA is a matrix of size m n. The following matrix norms are of particular interest.

While matrix multiplication does not commute the trace of a product of matrices does not depend on the order of multiplication. AB C A BC 4. The matrix B is the inverse of the matrix A and this is usually written as A1.

Properties of Scalar Multiplication. 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. Equally the matrix A is the inverse of the matrix B.

The matrix I is called the identity matrix and must have the same order as A. Let A B C be matrices and let c be a scalar. Properties of matrix multiplication.

Theorem 3 Algebraic Properties of Matrix Multiplication 1. 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.

Suppose are matrices and are scalars. For a square matrix A AI IA A where I is the identity matrix of the same order as A. 6 NM 1 1 1 2.

Properties of matrix operations The operations are as follows.

Properties Of Matrix Multiplication Article Khan Academy