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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:
 a11

a12
t

A 


 a1n
a21
a22
a2 n
am 1 

am 2 
.


amn 
Properties of Transposes
1. (At)t = A
Transpose of a transpose
2. (A + B) t = At+Bt
Transpose of a sum
3. (cA)t = c(At)
Transpose of a scalar
product
Transpose of a product
4. (AB)t = BtAt
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?