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+0mn=A

A+(-A) = 0mn

If cA=0mn then c=0 or A=0mn.

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)

AIn = A

ImA = A

assuming A is m by n and all operations are defined.

– Associative

– Left distributive

– Right Distributive

– Scalar Associative

– Multiplicative Identity

– 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 At (A transpose) to be the n by m matrix:

11 21 1

12 22 2

1 2

.

m

mt

n n mn

a a a

a a aA

a a a

Properties of Transposes

1. (At)t = A

2. (A + B) t = At+Bt

3. (cA)t = c(At)

4. (AB)t = BtAt

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-1A=In ?

• Does this always work?