Properties of Matrix Operations
Dr. Philippe B. Laval
Kennesaw State University
February 18, 2002
Abstract
This handout lists without proof the properties of the various matrix
operations. It also draws a parallel between matrix operations and operations on real numbers. Finally, it shows how these operations play a role
in solving equations involving matrices.
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Properties of Addition and Scalar Multiplication
Theorem 1 Let A, B 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).
The following properties hold:
1. A + B = B + A (commutative property)
2. (A + B) + C = A + (B + C) (associative property)
3. A + Omn = A (additive identity property)
4. A + (−A) = Omn (additive inverse property)
5. 1A = A
6. (cd) A = c (dA)
7. c (A + B) = cA + cB (distributive property)
8. (c + d) A = cA + dA (distributive property)
9. cA = Omn =⇒ c = 0 or A = Omn
These properties are very similar to the properties of addition and multiplication of real numbers. They allow us to solve matrix equations involving
addition and scalar multiplication the same way we solve equations involving
real numbers. Consider the example below of solving an equation with real
numbers and one with matrices. Notice the similarities.
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Example 2 Two similar equations
Real Numbers
2x + 3 = 7
2x + 3 + (−3) = 7 + (−3)
2x + 0 = 4
2x = 4
1
1
(2x) = 4
2 2
1
2 x=2
2
1x = 2
x=2
Matrices
cA + B = C
cA + B + (−B) = C + (−B)
cA + Omn = C − B
cA = C − B
1
1
(cA) = (C − B)
c
c 1
1
c A = (C − B)
c
c
1
1A = (C − B)
c
1
A = (C − B)
c
The steps are identical in both cases.
Remark 3 Some terminology we will revisit.
Theorem 4
1. Properties 2,3 and 4 say that the set of m×n matrices, Mm,n
together with matrix addition, is a group.
2. Properties 1, 2,3 and 4 say that the set of m × n matrices, Mm,n together
with matrix addition, is a commutative group or an Abelian group.
3. Properties 1 - 8 say that the set of m × n matrices, Mm,n together with
matrix addition and scalar multiplication, is a vector space.
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Properties of Matrix Multiplication and Inverse Matrices
Theorem 5 Let A, B and C be matrices with sizes such that the operations
below are defined, let In be the diagonal matrix of order n whose entries on
the main diagonal consist of 1 s (multiplicative identity matrix) and let c be a
scalar. The following properties hold:
1. (AB) C = A (BC) (associative property)
2. A (B + C) = AB + AC (distributive property)
3. (B + C) A = BA + CA (distributive property)
4. c (AB) = (cA) B
5. If A ∈ Mmn then AIn = A (In is the multiplicative identity)
6. If A ∈ Mmn then Im A = A (In is the multiplicative identity)
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7. If A is an invertible (also known as nonsingular) n × n matrix then it has
a unique inverse, denoted A−1 which satisfies AA−1 = A−1 A = In
Remark 6 Several important remarks should be made at this point.
1. Only square matrices can have an inverse; not every square matrix has an
inverse.
2. Matrix multiplication is not commutative. That is, in general AB = BA
3. If a matrix A is invertible, then it commutes with its inverse. In other
words, AA−1 = A−1 A. One application of this is that to check that a
matrix B is the inverse of a matrix A, it is enough to check that AB = I.
If this is true, it can be proven that BA must also be equal to I.
4. The cancellation law AC = BC ⇒ A = B does not hold in general (we
will see later that it holds if C is invertible). One consequence of this is
that it is possible for AB = 0 to happen, with A = 0 and B = 0. Tis can
never happen with real numbers. The principle of zero products says that
if ab = 0 then it must be that a = 0 or b = 0. This is often used when
solving equations. Remember that it does not always work for matrices.
Theorem 7 If A is an n × n invertible matrix, k is a positive integer and c is
a scalar then Ak , cA and At are invertible. Furthermore,
−1
1. A−1 is also nonsingular and A−1
=A
2. If B is also an n × n nonsingular matrix, then AB is nonsingular and
(AB)−1 = B −1 A−1
−1 −1 k
3. Ak
= A
4. (cA)−1 =
1 −1
A
c
Theorem 8 (Cancellation Property) If C is an invertible matrix, then the
following is true.
1. If AC = BC, then A = B. This is called the right cancellation property.
2. If CA = CB, then A = B. This is called the left cancellation property.
Theorem 9 (Solutions of a system of linear equations) If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution.
The solution is given by
x = A−1 b
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The Transpose of a Matrix
Definition 10 If A is an m × n matrix, then:
1. A is symmetric if At = A
2. A is skew-symmetric if At = −A
Theorem 11 Let A and B be matrices with sizes such that the operations below
are defined. The following properties hold:
t
1. (At ) = A
t
2. (A + B) = At + B t
3. (cA)t = cAt
4. If A is symmetric, then A is a square matrix.
t
5. (AB) = B t At
6. AAt and At A are symmetric matrices.
t
−1
7. (At ) = A−1
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Practice Problems
These problems are not computational. They deal more with your understanding of the various concepts.
1. Indicate whether each statement below is true or false. In either case,
justify your answer. When a statement is true, you can justify your answer
by either explaining why it is true, or citing a theorem or property. If the
statement is false, give a counter example. In addition, if the statement
is false, try to see what condition could be added to make it true.
(a) Every square matrix has an inverse.
(b) Every diagonal matrix has an inverse.
(c) 0nn is an invertible matrix.
(d) If A, B, C are square matrices then AC = BC ⇒ A = C.
(e) If a square matrix A is not symmetric, then At A is not symmetric.
(f) If A is nonsingular, then −A is also nonsingular.
(g) If a and B are nonsingular, then A + B is nonsingular.
(h) If A is singular, then Ax = b is consistent.
(i) If A is singular, then Ax = 0 is consistent.
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2. We have already said that it was possible to have two n × n matrices A
and B such that A = 0nn and B = 0nn and AB = 0nn . Give a condition
that either A or B or both would have to satisfy to have the equivalent of
the principle of zero products for matrices, that is AB = 0nn ⇒ A = 0nn
or B = 0nn .
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