Properties of Matrix Operations King Saud University

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

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

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.

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:

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

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?