# Matrix Operations

**THEOREM 2**

Let A be m × n and let B and C have sizes for which the
indicated sums and products are

defined.

a. A(BC) = (AB) C (associative law of
multiplication)

b. A (B + C) = AB + AC (left - distributive law)

c. (B + C)A = BA + CA (right-distributive law)

d. r(AB) = (rA)B = A (rB)

for any scalar r

e.
(identity for matrix multiplication)

**WARNINGS**

Properties above are analogous to properties of real numbers. But NOT ALL real
number

properties correspond to matrix properties.

1. It is not the case that AB always equal BA. (see
Example 7, page 114)

2. Even if AB = AC, then B may not equal C. (see Exercise 10, page 116)

3. It is possible for AB = 0 even if A ≠ 0 and B ≠ 0. (see Exercise 12, page
116)

**Powers of A**

EXAMPLE.

If A is m × n, the transpose of A is the n × m matrix,
denoted by A^{T}, whose columns are formed

from the corresponding rows of A.

**EXAMPLE.**

EXAMPLE: Let
Compute AB,
(AB)^{T}, A^{T}B^{T} and B^{T}A^{T}.

Solution.

**THEOREM 3**

Let A and B denote matrices whose sizes are appropriate for the following sums
and

products.

a. (AT) ^{T} =
A (I.e., the transpose of A^{T} is A)

b. (A + B)^{T} =
A^{T} B^{T}

c. For any scalar r, (rA)^{T} =
rA^{T}

d. (AB)^{T} =
B^{T}A^{T} (I.e. the transpose of a product of matrices equals the product of their

transposes in reverse order. )

EXAMPLE. Prove that (ABC)^{T}_________.

Solution. By Theorem 3d,

## 2.2 The Inverse of a Matrix

The inverse of a real number a is denoted by a ^{-1}. For example,
and

An n × n matrix A is said to be invertible if there is an n × n matrix C satisfying

CA = AC = I_{n}

where I_{n} is the n × n identity matrix. We call C the inverse of A .

**FACT** If A is invertible, then the inverse is unique.

Proof. Assume B and C are both inverses of A. Then

B = BI = B (_____) = ( ______) ______ = I______ = C.

So the inverse is unique since any two inverses coincide.

The inverse of A is usually denoted by A ^{-1}.

We have

AA ^{-1} = A ^{-1}A = I_{n} |

Not all n × n matrices are invertible. A matrix which is not invertible is
sometimes called a

singular matrix. An invertible matrix is called nonsingular matrix.

**Theorem 4
**Let

If ad - bc ≠ 0, then A is invertible and

If ad - bc = 0, then A is not invertible.

Assume A is any invertible matrix and we wish to solve Ax = b. Then

_____Ax= _____b and so

Ix= _______ or x= _______.

Suppose w is also a solution to Ax = b. Then Aw = b and

_____Aw = _____b which means w = A^{-1}b.

So, w = A^{-1}b, which is in fact the same solution.

We have proved the following result.

**Theorem 5**

If A is an invertible n × n matrix, then for each b in R^{n}, the equation Ax = b has
the unique

solution x = A^{-1}b.

**EXAMPLE. **Use the inverse of A

Solution. Matrix form of the linear system.

**Theorem 6** Suppose A and B are invertible. Then the
following results hold.

a. A ^{-1} is invertible and (A^{-1})^{-1} =
A (i.e. A is the inverse of A^{-1}).

b. AB is invertible and (AB)^{-1} =
B^{-1}A^{-1}

c. A^{T} is invertible and (A^{T})^{-1} = (A^{-1})^{T}

Partial proof of part b.

Similarly, one can show that (B^{-1}A^{-1})(AB) = I.

**Theorem 6,** part b can be generalized to three or
more invertible matrices.

(ABC)^{-1}=
__________

Earlier, we saw a formula for finding the inverse of a 2 × 2 invertible matrix.
How do we find the

inverse of an invertible n × n matrix? To answer this question, we first look at
elementary

matrices.

## Elementary Matrices

**Definition**

An elementary matrix is one that is obtained by performing a single elementary
row

operation on an identity matrix.

EXAMPLE. Let

and

E_{1}, E_{2}, and E_{3} are elementary matrices. Why?

Observe the following products and describe how these products can be obtained
by elementary

row operations on A.

If an elementary row operation is performed on an m × n matrix A, the resulting
matrix can be

written as EA, where the m × m matrix E is created by performing the same row
operations on I_{m}.

Elementary matrices are invertible because row operations are reversible. To
determine the

inverse of an elementary matrix E, determine the elementary row operation needed
to transform

E back into I and apply this operation to I to find the inverse.

For example,

**Example.** Let

. Then

Then multiplying on the right by A^{-1}, we get

E_{3}E_{2}E_{1}A______= I_{3}______.

So

**E _{3}E_{2}E_{1}I_{3} = A^{-1}**

**The elementary row operations that row reduce A to I _{n} are the same elementary
row
operations that transform I_{n} into A^{-1}.**