2.5 Non{singular matrices

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DEFINITION 2.5.1 (Non{singular matrix)

A square matrix A 2 Mn£n(F) is called non{singular or invertible if

there exists a matrix B 2 Mn£n(F) such that

AB = In = BA:

Any matrix B with the above property is called an inverse of A. If A does

not have an inverse, A is called singular.

2.5. NON{SINGULAR MATRICES 37

THEOREM 2.5.1 (Inverses are unique)

If A has inverses B and C, then B = C.

Proof. Let B and C be inverses of A. Then AB = In = BA and AC =

In = CA. Then B(AC) = BIn = B and (BA)C = InC = C. Hence because

B(AC) = (BA)C, we deduce that B = C.

REMARK 2.5.1 If A has an inverse, it is denoted by A¡1. So

AA¡1 = In = A¡1A:

Also if A is non{singular, it follows that A¡1 is also non{singular and

(A¡1)¡1 = A:

THEOREM 2.5.2 If A and B are non{singular matrices of the same size,

then so is AB. Moreover

(AB)¡1 = B¡1A¡1:

Proof.

(AB)(B¡1A¡1) = A(BB¡1)A¡1 = AInA¡1 = AA¡1 = In:

Similarly

(B¡1A¡1)(AB) = In:

REMARK 2.5.2 The above result generalizes to a product of m non{

singular matrices: If A1; : : : ;Am are non{singular n £ n matrices, then the

product A1 : : :Am is also non{singular. Moreover

(A1 : : :Am)¡1 = A¡1

m : : :A¡1

1 :

(Thus the inverse of the product equals the product of the inverses in the

reverse order.)

EXAMPLE 2.5.1 If A and B are n £ n matrices satisfying A2 = B2 =

(AB)2 = In, prove that AB = BA.

Solution. Assume A2 = B2 = (AB)2 = In. Then A; B; AB are non{

singular and A¡1 = A; B¡1 = B; (AB)¡1 = AB.

But (AB)¡1 = B¡1A¡1 and hence AB = BA.

38 CHAPTER 2. MATRICES

EXAMPLE 2.5.2 A =

·

1 2

4 8

¸

is singular. For suppose B =

·

a b

c d

¸

is an inverse of A. Then the equation AB = I2 gives

·

1 2

4 8

¸ ·

a b

c d

¸

=

·

1 0

0 1

¸

and equating the corresponding elements of column 1 of both sides gives the

system

a + 2c = 1

4a + 8c = 0

which is clearly inconsistent.

THEOREM 2.5.3 Let A =

·

a b

c d

¸

and ¢ = ad ¡ bc 6= 0. Then A is

non{singular. Also

A¡1 = ¢¡1

·

d ¡b

¡c a

¸

:

REMARK 2.5.3 The expression ad ¡ bc is called the determinant of A

and is denoted by the symbols detA or

¯¯¯¯

a b

c d

¯¯¯¯

.

Proof. Verify that the matrix B = ¢¡1

·

d ¡b

¡c a

¸

satis¯es the equation

AB = I2 = BA.

EXAMPLE 2.5.3 Let

A =

2

4

0 1 0

0 0 1

5 0 0

3

5 :

Verify that A3 = 5I3, deduce that A is non{singular and ¯nd A¡1.

Solution. After verifying that A3 = 5I3, we notice that

A

µ

1

5

A2

= I3 =

µ

1

5

A2

A:

Hence A is non{singular and A¡1 = 1

5A2.

2.5. NON{SINGULAR MATRICES 39

THEOREM 2.5.4 If the coe±cient matrix A of a system of n equations

in n unknowns is non{singular, then the system AX = B has the unique

solution X = A¡1B.

Proof. Assume that A¡1 exists.

1. (Uniqueness.) Assume that AX = B. Then

(A¡1A)X = A¡1B;

InX = A¡1B;

X = A¡1B:

2. (Existence.) Let X = A¡1B. Then

AX = A(A¡1B) = (AA¡1)B = InB = B:

THEOREM 2.5.5 (Cramer's rule for 2 equations in 2 unknowns)

The system

ax + by = e

cx + dy = f

has a unique solution if ¢ =

¯¯¯¯

a b

c d

¯¯¯¯

6= 0, namely

x =

¢1

¢

; y =

¢2

¢

;

where

¢1 =

¯¯¯¯

e b

f d

¯¯¯¯

and ¢2 =

¯¯¯¯

a e

c f

¯¯¯¯

:

Proof. Suppose ¢ 6= 0. Then A =

·

a b

c d

¸

has inverse

A¡1 = ¢¡1

·

d ¡b

¡c a

¸

and we know that the system

A

·

x

y

¸

=

·

e

f

¸

40 CHAPTER 2. MATRICES

has the unique solution

·

x

y

¸

= A¡1

·

e

f

¸

=

1

¢

·

d ¡b

¡c a

¸ ·

e

f

¸

=

1

¢

·

de ¡ bf

¡ce + af

¸

=

1

¢

·

¢1

¢2

¸

=

·

¢1=¢

¢2=¢

¸

:

Hence x = ¢1=¢; y = ¢2=¢.

COROLLARY 2.5.1 The homogeneous system

ax + by = 0

cx + dy = 0

has only the trivial solution if ¢ =

¯¯¯¯

a b

c d

¯¯¯¯

6= 0.

EXAMPLE 2.5.4 The system

7x + 8y = 100

2x ¡ 9y = 10

has the unique solution x = ¢1=¢; y = ¢2=¢, where

¢ =

¯¯¯¯

7 8

2 ¡9

¯¯¯¯

= ¡79; ¢1 =

¯¯¯¯

100 8

10 ¡9

¯¯¯¯

= ¡980; ¢2 =

¯¯¯¯

7 100

2 10

¯¯¯¯

= ¡130:

So x = 980

79 and y = 130

79 .

THEOREM 2.5.6 Let A be a square matrix. If A is non{singular, the

homogeneous system AX = 0 has only the trivial solution. Equivalently,

if the homogenous system AX = 0 has a non{trivial solution, then A is

singular.

Proof. If A is non{singular and AX = 0, then X = A¡10 = 0.

REMARK 2.5.4 If A¤1; : : : ;A¤n denote the columns of A, then the equa-

tion

AX = x1A¤1 + : : : + xnA¤n

holds. Consequently theorem 2.5.6 tells us that if there exist scalars x1; : : : ; xn,

not all zero, such that

x1A¤1 + : : : + xnA¤n = 0;

2.5. NON{SINGULAR MATRICES 41

that is, if the columns of A are linearly dependent, then A is singular. An

equivalent way of saying that the columns of A are linearly dependent is that

one of the columns of A is expressible as a sum of certain scalar multiples

of the remaining columns of A; that is one column is a linear combination

of the remaining columns.

EXAMPLE 2.5.5

A =

2

4

1 2 3

1 0 1

3 4 7

3

5

is singular. For it can be veri¯ed that A has reduced row{echelon form

2

4

1 0 1

0 1 1

0 0 0

3

5

and consequently AX = 0 has a non{trivial solution x = ¡1; y = ¡1; z = 1.

REMARK 2.5.5 More generally, if A is row{equivalent to a matrix con-

taining a zero row, then A is singular. For then the homogeneous system

AX = 0 has a non{trivial solution.

An important class of non{singular matrices is that of the elementary

row matrices.

DEFINITION 2.5.2 (Elementary row matrices) There are three types,

Eij ; Ei(t); Eij(t), corresponding to the three kinds of elementary row oper-

ation:

1. Eij ; (i 6= j) is obtained from the identity matrix In by interchanging

rows i and j.

2. Ei(t); (t 6= 0) is obtained by multiplying the i{th row of In by t.

3. Eij(t); (i 6= j) is obtained from In by adding t times the j{th row of

In to the i{th row.

EXAMPLE 2.5.6 (n = 3.)

E23 =

2

4

1 0 0

0 0 1

0 1 0

3

5 ; E2(¡1) =

2

4

1 0 0

0 ¡1 0

0 0 1

3

5 ; E23(¡1) =

2

4

1 0 0

0 1 ¡1

0 0 1

3

5 :

42 CHAPTER 2. MATRICES

The elementary row matrices have the following distinguishing property:

THEOREM 2.5.7 If a matrix A is pre{multiplied by an elementary row{

matrix, the resulting matrix is the one obtained by performing the corre-

sponding elementary row{operation on A.

EXAMPLE 2.5.7

E23

2

4

a b

c d

e f

3

5 =

2

4

1 0 0

0 0 1

0 1 0

3

5

2

4

a b

c d

e f

3

5 =

2

4

a b

e f

c d

3

5 :

COROLLARY 2.5.2 The three types of elementary row{matrices are non{

singular. Indeed

1. E¡1

ij = Eij ;

2. E¡1

i (t) = Ei(t¡1);

3. (Eij(t))¡1 = Eij(¡t).

Proof. Taking A = In in the above theorem, we deduce the following

equations:

EijEij = In

Ei(t)Ei(t¡1) = In = Ei(t¡1)Ei(t) if t 6= 0

Eij(t)Eij(¡t) = In = Eij(¡t)Eij(t):

EXAMPLE 2.5.8 Find the 3 £ 3 matrix A = E3(5)E23(2)E12 explicitly.

Also ¯nd A¡1.

Solution.

A = E3(5)E23(2)

2

4

0 1 0

1 0 0

0 0 1

3

5 = E3(5)

2

4

0 1 0

1 0 2

0 0 1

3

5 =

2

4

0 1 0

1 0 2

0 0 5

3

5 :

To ¯nd A¡1, we have

A¡1 = (E3(5)E23(2)E12)¡1

= E¡1

12 (E23(2))¡1 (E3(5))¡1

= E12E23(¡2)E3(5¡1)

2.5. NON{SINGULAR MATRICES 43

= E12E23(¡2)

2

4

1 0 0

0 1 0

0 0 1

5

3

5

= E12

2

4

1 0 0

0 1 ¡2

5

0 0 1

5

3

5 =

2

4

0 1 ¡2

5

1 0 0

0 0 1

5

3

5 :

REMARK 2.5.6 Recall that A and B are row{equivalent if B is obtained

from A by a sequence of elementary row operations. If E1; : : : ;Er are the

respective corresponding elementary row matrices, then

B = Er (: : : (E2(E1A)) : : :) = (Er : : :E1)A = PA;

where P = Er : : :E1 is non{singular. Conversely if B = PA, where P is

non{singular, then A is row{equivalent to B. For as we shall now see, P is

in fact a product of elementary row matrices.

THEOREM 2.5.8 Let A be non{singular n £ n matrix. Then

(i) A is row{equivalent to In,

(ii) A is a product of elementary row matrices.

Proof. Assume that A is non{singular and let B be the reduced row{echelon

form of A. Then B has no zero rows, for otherwise the equation AX = 0

would have a non{trivial solution. Consequently B = In.

It follows that there exist elementary row matrices E1; : : : ;Er such that

Er (: : : (E1A) : : :) = B = In and hence A = E¡1

1 : : :E¡1

r , a product of

elementary row matrices.

THEOREM 2.5.9 Let A be n £ n and suppose that A is row{equivalent

to In. Then A is non{singular and A¡1 can be found by performing the

same sequence of elementary row operations on In as were used to convert

A to In.

Proof. Suppose that Er : : :E1A = In. In other words BA = In, where

B = Er : : :E1 is non{singular. Then B¡1(BA) = B¡1In and so A = B¡1,

which is non{singular.

Also A¡1 =

¡

B¡1

¢

¡1 = B = Er ((: : : (E1In) : : :), which shows that A¡1

is obtained from In by performing the same sequence of elementary row

operations as were used to convert A to In.

44 CHAPTER 2. MATRICES

REMARK 2.5.7 It follows from theorem 2.5.9 that if A is singular, then

A is row{equivalent to a matrix whose last row is zero.

EXAMPLE 2.5.9 Show that A =

·

1 2

1 1

¸

is non{singular, ¯nd A¡1 and

express A as a product of elementary row matrices.

Solution. We form the partitioned matrix [AjI2] which consists of A followed

by I2. Then any sequence of elementary row operations which reduces A to

I2 will reduce I2 to A¡1. Here

[AjI2] =

·

1 2 1 0

1 1 0 1

¸

R2 ! R2 ¡ R1

·

1 2 1 0

0 ¡1 ¡1 1

¸

R2 ! (¡1)R2

·

1 2 1 0

0 1 1 ¡1

¸

R1 ! R1 ¡ 2R2

·

1 0 ¡1 2

0 1 1 ¡1

¸

:

Hence A is row{equivalent to I2 and A is non{singular. Also

A¡1 =

·

¡1 2

1 ¡1

¸

:

We also observe that

E12(¡2)E2(¡1)E21(¡1)A = I2:

Hence

A¡1 = E12(¡2)E2(¡1)E21(¡1)

A = E21(1)E2(¡1)E12(2):

The next result is the converse of Theorem 2.5.6 and is useful for proving

the non{singularity of certain types of matrices.

THEOREM 2.5.10 Let A be an n £ n matrix with the property that

the homogeneous system AX = 0 has only the trivial solution. Then A is

non{singular. Equivalently, if A is singular, then the homogeneous system

AX = 0 has a non{trivial solution.

2.5. NON{SINGULAR MATRICES 45

Proof. If A is n £ n and the homogeneous system AX = 0 has only the

trivial solution, then it follows that the reduced row{echelon form B of A

cannot have zero rows and must therefore be In. Hence A is non{singular.

COROLLARY 2.5.3 Suppose that A and B are n £ n and AB = In.

Then BA = In.

Proof. Let AB = In, where A and B are n £ n. We ¯rst show that B

is non{singular. Assume BX = 0. Then A(BX) = A0 = 0, so (AB)X =

0; InX = 0 and hence X = 0.

Then from AB = In we deduce (AB)B¡1 = InB¡1 and hence A = B¡1.

The equation BB¡1 = In then gives BA = In.

Before we give the next example of the above criterion for non-singularity,

we introduce an important matrix operation.

DEFINITION 2.5.3 (The transpose of a matrix) Let A be an m£n

matrix. Then At, the transpose of A, is the matrix obtained by interchanging

the rows and columns of A. In other words if A = [aij ], then

¡

At

¢

ji = aij .

Consequently At is n £ m.

The transpose operation has the following properties:

1.

¡

At

¢t = A;

2. (A § B)t = At § Bt if A and B are m £ n;

3. (sA)t = sAt if s is a scalar;

4. (AB)t = BtAt if A is m £ n and B is n £ p;

5. If A is non{singular, then At is also non{singular and

¡

At¢

¡1

=

¡

A¡1¢t

;

6. XtX = x2

1 + : : : + x2

n if X = [x1; : : : ; xn]t is a column vector.

We prove only the fourth property. First check that both (AB)t and BtAt

have the same size (p £ m). Moreover, corresponding elements of both

matrices are equal. For if A = [aij ] and B = [bjk], we have

¡

(AB)t¢

ki = (AB)ik

=

Xn

j=1

aijbjk

46 CHAPTER 2. MATRICES

=

Xn

j=1

¡

Bt¢

kj

¡

At¢

ji

=

¡

BtAt¢

ki :

There are two important classes of matrices that can be de¯ned concisely

in terms of the transpose operation.

DEFINITION 2.5.4 (Symmetric matrix) A real matrix A is called sym-

metric if At = A. In other words A is square (n £ n say) and aji = aij for

all 1 · i · n; 1 · j · n. Hence

A =

·

a b

b c

¸

is a general 2 £ 2 symmetric matrix.

DEFINITION 2.5.5 (Skew{symmetric matrix) A real matrix A is called

skew{symmetric if At = ¡A. In other words A is square (n £ n say) and

aji = ¡aij for all 1 · i · n; 1 · j · n.

REMARK 2.5.8 Taking i = j in the de¯nition of skew{symmetric matrix

gives aii = ¡aii and so aii = 0. Hence

A =

·

0 b

¡b 0

¸

is a general 2 £ 2 skew{symmetric matrix.

We can now state a second application of the above criterion for non{

singularity.

COROLLARY 2.5.4 Let B be an n £ n skew{symmetric matrix. Then

A = In ¡ B is non{singular.

Proof. Let A = In ¡ B, where Bt = ¡B. By Theorem 2.5.10 it su±ces to

show that AX = 0 implies X = 0.

We have (In ¡ B)X = 0, so X = BX. Hence XtX = XtBX.

Taking transposes of both sides gives

(XtBX)t = (XtX)t

XtBt(Xt)t = Xt(Xt)t

Xt(¡B)X = XtX

¡XtBX = XtX = XtBX:

Hence XtX = ¡XtX and XtX = 0. But if X = [x1; : : : ; xn]t, then XtX =

x2

1 + : : : + x2

n = 0 and hence x1 = 0; : : : ; xn = 0.

2.6. LEAST SQUARES SOLUTION OF EQUATIONS 47

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