Download presentation
Presentation is loading. Please wait.
Published byBrenda Stokes Modified over 9 years ago
1
Properties of Matrix Operations King Saud University
2
Properties of Matrix Addition A+B = B+A A+(B+C)=(A+B)+C (cd)A = c(dA) 1A=A c(A+B) = cA+cB (c+d)A = cA +dA A+0 mn =A A+(-A) = 0 mn If cA=0 mn then c=0 or A=0 mn. Commutative Associative Scalar Associative Scalar identity Scalar distributive 1 Scalar distributive 2 Additive identity Additive Inverse Scalar cancellation property
3
Properties of Matrix Multiplication A(BC) = (AB)C A(B+C) = AB +AC (A+B)C = AC+BC c(AB) = (cA)B=A(cB) AI n = A I m A = A assuming A is m by n and all operations are defined. –Associative –Left distributive –Right Distributive –Scalar Associative –Multiplicative Identity
4
Using Properties to Prove Theorems Using these properties we can prove the following theorem (which we have already been assuming). Theorem: For a system of linear equations in n variables, precisely one of the following is true: 1. The system has exactly one solution. 2. The system has an infinite number of solutions. 3. The system has no solutions.
5
The Transpose of a Matrix We will find it useful at times to talk about the transpose of a matrix. Given an m by n matrix A, we define A t ( A transpose ) to be the n by m matrix:
6
Properties of Transposes 1. (A t ) t = A 2. (A + B) t = A t +B t 3. (cA) t = c(A t ) 4. (AB) t = B t A t Transpose of a transpose Transpose of a sum Transpose of a scalar product Transpose of a product
7
What about Mult. Inverses For an n by n matrix A, can we find an n by n matrix A -1 so that AA -1 =A -1 A=I n ? Does this always work?
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.