1. Matrices 3 1

2. Matrix Algebra In this lecture we compare matrix addition and
multiplication with the addition and multiplication
of real numbers.
We shall see that there are similarities, but one has
to be more careful with matrix operations, as they
lack some of the properties we usually take for
granted.
We look at commutativity, zero matrices, identity
matrices, and inverses. 2

3. Commutativity Matrix addition is commutative: if A + B can be
calculated then B + A also exists and
A + B = B + A.
But multiplication is not commutative:
It may be that AB is defined but BA is
undefined. (Let A be 2 × 3 and B be 3 × 3.)
It may be that AB and BA are both defined, but
different sizes. (Let A be 2 × 3 and B be 3 × 2.) 3

4. Commutativity It may be that AB and BA are both defined,
and the same size, but give different matrices.
For example:
whereas 4

5. Zero Matrices If x is any real number, then 0+x = x and x+0 = x.
A zero matrix is a matrix all of whose entries are
zero. So
is the 2 × 3 zero matrix, and of course one can
have a zero matrix of any size. 5

6. Zero Matrices If A is any matrix and O is the zero matrix with
the same size as A, then
O + A = A = A + O
For example 6

7. Identity Matrices If x is any real number, then 1·x = x and x·1 = x.
For matrices we have a similar property, provided
we restrict ourselves to square matrices, so that
multiplication is always defined and delivers a
product of the same size.
A square matrix is one that has as many rows as columns, and so is n×n.
The n×n identity matrix In has 1’s on the diagonal
from top left to bottom right, and 0’s elsewhere. 7

8. Identity Matrices So If A is any n × n matrix then InA = A = AIn
For example 8

9. The Idea Of An Inverse
If x is any real number, then x+(-x) = 0 = -x+x.
We say that -x is the additive inverse of x.
Similarly, for any matrix A, we have
A + (-1)A = O = (-1)A + A
For example
Thus every matrix has an additive inverse. 9

10. The Idea Of An Inverse If x is any nonzero real number, then
We say that (or x-1) is the multiplicative inverse of x.
Note that not every real number has a multiplicative
inverse, just the nonzero numbers.
If we hope to find a similar property for matrices, we
should once again restrict ourselves to square matrices,
so that multiplication is always defined.
Do some matrices have multiplicative inverses?
Which ones? And how do we find the multiplicative
inverse of a matrix, assuming it has one? 10

11. Invertible Matrices Let A be any n × n matrix. Then A is invertible
iff the rank of A is n. (We shall not prove this
fact.)
To put it differently, A has a multiplicative
inverse A-1 iff no rows consisting just of 0’s are
produced when we use elementary row
transformations to reduce A to echelon form. 11

12. Invertible Matrices For example, let A =
Then A is invertible, since R2 ⟶ 4R2 - R1 gives
which is in echelon form and has no rows
consisting entirely of zeros. 12

13. Finding Inverses Suppose we have a matrix like A =
which has an inverse. How do we find the inverse?
The most obvious method is to write 13

14. Finding Inverses
and to solve the corresponding 2 systems of linear equations for the unknowns x1, y1 and x2, y2:
2x1 + 2y1 = 1
x1 + y1 = 0
and
2x2 + 2y2 = 0
x2 + y2 = 1 14

15. The Gauss-Jordan Method
The Gauss-Jordan method streamlines the process
of finding A-1 for any square matrix A that is
invertible.
1. Write down the matrix A and on its right an
identity matrix of the same size.
2. Perform elementary row-transformations on the
lefthand matrix (which starts off as A) so as to
change it into an identity matrix, and at the
same time perform the same elementary
transformations on the righthand matrix (which
starts of as the identity matrix). 15

16. The Gauss-Jordan Method
When the lefthand matrix has changed into
the identity matrix, the righthand matrix will
be the inverse of A.
If A is not invertible, then the lefthand matrix
will not change into an identity matrix but
instead will have a row consisting entirely of
zeros. 16

17. Example We find A-1 for A = Step 1:
Step 2: Now R1 ⟶ R1, followed by R2 ⟶ 2 R2 - R1,
and finishing with R1 ⟶ R1 - R2 give 17

18. Example
so that
A-1 =
We can check this by multiplying: 18

19. What Use Are Inverse Matrices?
Consider the system of linear equations
2x + 2y = 4
x + y = 3
which can be written as a matrix equation 19

20. What Use Are Inverse Matrices?
If A-1 exists, then a matrix equation AX = B has
the solution X = A-1 B, since we may multiply
from the left by A-1 on both sides of AX = B to
get A-1 AX = A-1 B, and A-1 AX = IX = X.
In the case of our example we know A-1 and so
so that x = -2 and y = 4. 20