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.

1 Properties of Addition and Scalar Multiplica- tion

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 =0or 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.

1 Example 2 Two similar equations

Real Numbers Matrices 2x +3=7 cA + B = C 2x +3+(−3) = 7 + (−3) cA + B +(−B)=C +(−B) 2x +0=4 cA + Omn = C − B 2x =4 cA = C − B 1 1 1 1 (2x)= 4 (cA)= (C − B) 2 2 c c 1 1 1 2 x =2 c A = (C − B) 2 c c 1 1x =2 1A = (C − B) c 1 x =2 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.

2 Properties of Matrix Multiplication and In- verse 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 1s (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 ImA = A (In is the multiplicative identity)

2 7. If A is an invertible (also known as nonsingular) n × n matrix then it has −1 −1 −1 a unique inverse, denoted A which satisﬁes AA = A 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−1A. 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 =0to happen, with A =0and B =0.Tiscan never happen with real numbers. The principle of zero products says that if ab =0then it must be that a =0or 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 ascalarthenAk, 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−1A−1

−1 k 3. Ak = A−1 1 (cA)−1 = A−1 4. c

Theorem 8 (Cancellation Property) If C is an invertible matrix, then the following is true.

1. If AC = BC,thenA = B. This is called the right cancellation property. 2. If CA = CB,thenA = 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−1b

3 3 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 + Bt t 3. (cA) = cAt 4. If A is symmetric, then A is a square matrix. t 5. (AB) = BtAt 6. AAt and AtA are symmetric matrices. −1 t 7. (At) = A−1

4 Practice Problems

These problems are not computational. They deal more with your understand- ing 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 AtA 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.

4 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.