2.1 Matrix arithmetic

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A matrix over a ¯eld F is a rectangular array of elements from F. The sym-

bol Mm£n(F) denotes the collection of all m£n matrices over F. Matrices

will usually be denoted by capital letters and the equation A = [aij ] means

that the element in the i{th row and j{th column of the matrix A equals

aij . It is also occasionally convenient to write aij = (A)ij . For the present,

all matrices will have rational entries, unless otherwise stated.

EXAMPLE 2.1.1 The formula aij = 1=(i + j) for 1 · i · 3; 1 · j · 4

de¯nes a 3 £ 4 matrix A = [aij ], namely

A =

2

66664

1

2

1

3

1

4

1

5

1

3

1

4

1

5

1

6

1

4

1

5

1

6

1

7

3

77775:

DEFINITION 2.1.1 (Equality of matrices) Matrices A and B are said

to be equal if A and B have the same size and corresponding elements are

equal; that is A and B 2 Mm£n(F) and A = [aij ]; B = [bij ], with aij = bij

for 1 · i · m; 1 · j · n.

DEFINITION 2.1.2 (Addition of matrices) Let A = [aij ] and B =

[bij ] be of the same size. Then A + B is the matrix obtained by adding

corresponding elements of A and B; that is

A + B = [aij ] + [bij ] = [aij + bij ]:

23

24 CHAPTER 2. MATRICES

DEFINITION 2.1.3 (Scalar multiple of a matrix) Let A = [aij ] and

t 2 F (that is t is a scalar). Then tA is the matrix obtained by multiplying

all elements of A by t; that is

tA = t[aij ] = [taij ]:

DEFINITION 2.1.4 (Additive inverse of a matrix) Let A = [aij ] .

Then ¡A is the matrix obtained by replacing the elements of A by their

additive inverses; that is

¡A = ¡[aij ] = [¡aij ]:

DEFINITION 2.1.5 (Subtraction of matrices) Matrix subtraction is

de¯ned for two matrices A = [aij ] and B = [bij ] of the same size, in the

usual way; that is

A ¡ B = [aij ] ¡ [bij ] = [aij ¡ bij ]:

DEFINITION 2.1.6 (The zero matrix) For each m; n the matrix in

Mm£n(F), all of whose elements are zero, is called the zero matrix (of size

m £ n) and is denoted by the symbol 0.

The matrix operations of addition, scalar multiplication, additive inverse

and subtraction satisfy the usual laws of arithmetic. (In what follows, s and

t will be arbitrary scalars and A; B; C are matrices of the same size.)

1. (A + B) + C = A + (B + C);

2. A + B = B + A;

3. 0 + A = A;

4. A + (¡A) = 0;

5. (s + t)A = sA + tA, (s ¡ t)A = sA ¡ tA;

6. t(A + B) = tA + tB, t(A ¡ B) = tA ¡ tB;

7. s(tA) = (st)A;

8. 1A = A, 0A = 0, (¡1)A = ¡A;

9. tA = 0 ) t = 0 or A = 0.

Other similar properties will be used when needed.

2.1. MATRIX ARITHMETIC 25

DEFINITION 2.1.7 (Matrix product) Let A = [aij ] be a matrix of

size m £ n and B = [bjk] be a matrix of size n £ p; (that is the number

of columns of A equals the number of rows of B). Then AB is the m £ p

matrix C = [cik] whose (i; k){th element is de¯ned by the formula

cik =

Xn

j=1

aijbjk = ai1b1k + ¢ ¢ ¢ + ainbnk:

EXAMPLE 2.1.2

1.

·

1 2

3 4

¸ ·

5 6

7 8

¸

=

·

1 £ 5 + 2 £ 7 1 £ 6 + 2 £ 8

3 £ 5 + 4 £ 7 3 £ 6 + 4 £ 8

¸

=

·

19 22

43 50

¸

;

2.

·

5 6

7 8

¸ ·

1 2

3 4

¸

=

·

23 34

31 46

¸

6=

·

1 2

3 4

¸ ·

5 6

7 8

¸

;

3.

·

1

2

¸ £

3 4

¤

=

·

3 4

6 8

¸

;

4.

£

3 4

¤ ·

1

2

¸

=

£

11

¤

;

5.

·

1 ¡1

1 ¡1

¸ ·

1 ¡1

1 ¡1

¸

=

·

0 0

0 0

¸

.

Matrix multiplication obeys many of the familiar laws of arithmetic apart

from the commutative law.

1. (AB)C = A(BC) if A; B; C are m £ n; n £ p; p £ q, respectively;

2. t(AB) = (tA)B = A(tB), A(¡B) = (¡A)B = ¡(AB);

3. (A + B)C = AC + BC if A and B are m £ n and C is n £ p;

4. D(A + B) = DA + DB if A and B are m £ n and D is p £ m.

We prove the associative law only:

First observe that (AB)C and A(BC) are both of size m £ q.

Let A = [aij ]; B = [bjk]; C = [ckl]. Then

((AB)C)il =

Xp

k=1

(AB)ikckl =

Xp

k=1

0

@

Xn

j=1

aijbjk

1

Ackl

=

Xp

k=1

Xn

j=1

aijbjkckl:

26 CHAPTER 2. MATRICES

Similarly

(A(BC))il =

Xn

j=1

Xp

k=1

aijbjkckl:

However the double summations are equal. For sums of the form

Xn

j=1

Xp

k=1

djk and

Xp

k=1

Xn

j=1

djk

represent the sum of the np elements of the rectangular array [djk], by rows

and by columns, respectively. Consequently

((AB)C)il = (A(BC))il

for 1 · i · m; 1 · l · q. Hence (AB)C = A(BC).

The system of m linear equations in n unknowns

a11x1 + a12x2 + ¢ ¢ ¢ + a1nxn = b1

a21x1 + a22x2 + ¢ ¢ ¢ + a2nxn = b2

...

am1x1 + am2x2 + ¢ ¢ ¢ + amnxn = bm

is equivalent to a single matrix equation

2

6664

a11 a12 ¢ ¢ ¢ a1n

a21 a22 ¢ ¢ ¢ a2n

...

...

am1 am2 ¢ ¢ ¢ amn

3

7775

2

6664

x1

x2

...

xn

3

7775

=

2

6664

b1

b2

...

bm

3

7775

;

that is AX = B, where A = [aij ] is the coe±cient matrix of the system,

X =

2

6664

x1

x2

...

xn

3

7775

is the vector of unknowns and B =

2

6664

b1

b2

...

bm

3

7775

is the vector of

constants.

Another useful matrix equation equivalent to the above system of linear

equations is

x1

2

6664

a11

a21

...

am1

3

7775+

x

2 2

6664

a12

a22

...

am2

3

7775

+ ¢ ¢ ¢ + xn

2

6664

a1n

a2n

...

amn

3

7775

=

2

6664

b1

b2

...

bm

3

7775

:

2.2. LINEAR TRANSFORMATIONS 27

EXAMPLE 2.1.3 The system

x + y + z = 1

x ¡ y + z = 0:

is equivalent to the matrix equation

·

1 1 1

1 ¡1 1

¸ 2

4

x

y

z

3

5 =

·

1

0

¸

and to the equation

x

·

1

1

¸

+ y

·

1

¡1

¸

+ z

·

1

1

¸

=

·

1

0

¸

:

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