Rank of a matrix. Theorem of Kronecker-Capelli

1.

Rank of a matrix. Theorem of Kronecker-Capelli
Consider a system of n linear equations with n variables
(x1, x2, …, xn):
a x a x ... a x b (*)
11 1
12
2
1n
n
1
a x a x ... a x b
21 1
22 2
2n n
2
..........................................
a n1 x1 a n 2 x 2 ... a nn x n bn
Consider
a11
a
A(n; n) 21
...
a
n1
a12
a 22
...
an2
... a1n
... a 2 n
... ...
... a nn
x1
x
X ( n;1) 2
x
n
b1
b
B (n;1) 2 .
b
n

2.

Then the system (*) is written in a matrix
representation: A(n; n) X(n; 1) = B(n; 1)
X = A-1 B.
A matrix A(n; n) is called regular if its
determinant is not equal to zero, i.e.
a11
a12
... a1n
a 21
a 22
... a 2 n
...
...
...
a n1
an2
... a nn
...
0.
A matrix A-1(n; n) is called inverse to a matrix
A(n; n) if the product
A(n; n) A-1(n; n)= A-1(n; n) A(n; n) = E(n; n),

3.

A11
1 A12
1
A
...
A
1n
A21
A22
...
A2 n
An1
... An 2
.
... ...
... Ann
...
Example . Find the inverse matrix to the matrix
3 1 0
A 2 1 1 .
2 1 4
3
1) 2
2
1 0
1
1 12 2 0 0 8 3 5 0.
1 4
2) А11 = 5; A12 = 10; A13 = 0; A21 = 4; A22 = 12; A23 = 1; A31 = -1;
A32 = -3; A33 = 1.
5 10 0
5 4 1
5 4 1
1
T
1
3) A* 4 12 1 . 4) A * 10 12 3 . 5) A 10 12 3 .
5
1 3 1
0 1 1
0 1 1

4.

3 1 0 5 4 1
5 0 0 1 0 0
1
1
1
A A 2 1 1 10 12 3 0 5 0 0 1 0 .
5
0 1 1 5 0 0 5 0 0 1
2
1
4
Rank of a matrix.
Consider a matrix of the dimension m n :
a11
a
A( m; n) 21
...
a
m1
a12
a 22
...
am2
... a1n
... a 2 n
... ...
... a mn
The rank of a matrix A is the greatest order of its nonequal to zero minors. The rank of a matrix is denoted by
Rank A or r(A).
Theorem. If there is a non-equal to zero minor of the rth order in a matrix A and all its bordering minors of the
r+1-th order are equal to zero then the rank of A is
equal to r, i.e. r(A)=r.

5.

Theorem. The rank of a matrix doesn’t change if:
a) All the rows are replaced by the corresponding
columns and vice versa;
b) Replace two arbitrary rows (columns);
c) Multiply (divide) each element of a row
(column) on the same non-zero number;
d) Add to (subtract from) elements of a row
(column) the corresponding elements of any other
row (column) multiplied on the same non-zero
number.

6.

Theorem of Kronecker-Capelli. A system of
linear equations is consistent if the rank of the
basic matrix A equals the rank of the extended
matrix C, i.e. Rank A = Rank C. Moreover:
•1) If Rank A = Rank C = n (where n is the
number of variables in the system) then the
system has a unique solution.
•2) If Rank A = Rank C < n then the system has
infinitely many solutions.

7.

Solving a system of linear equations
by the Gauss method
a11 x a12 y a13 z a14u a15
a x a y a z a u a
21
22
23
24
25
a31 x a32 y a33 z a34u a35
a 41 x a 42 y a 43 z a 44u a 45
(a)
(b)
(c )
(d )
Suppose that а11 0 (if а11 = 0 then we change the order of
equations by choosing as the first equation such an equation in
which the coefficient of х is not equal to zero).
I step: divide the equation (а) on а11, then multiply the obtained
equation on а21 and subtract from (b); further multiply (а)/a11 on
а31 and subtract from (с); at last, miltiply (а)/a11 on а41 and
subtract from (d).

8.

x b12 y b13 z b14u b15 (e)
b22 y b23 z b24u b25 ( f )
b32 y b33 z b34u b35 ( g )
b42 y b43 z b44u b45 (i )
where bij are obtained from aij by the following formulas:
b1j = a1j / a11 (j = 2, 3, 4, 5);
bij = aij – ai1 b1j (i = 2, 3, 4; j = 2, 3, 4, 5).
II step: do the same actions with (f), (g), (i) (as with (a), (b),
(c), (d)) and etc.
As a final result the initial system will be transformed to a socalled step form:
x b12 y b13 z b14 u b15
y c 23 z c 24 u c 25
z d 34 u d 35
u e45

9.

Example 1.
3 x 2 y z 5
x y z 0
4 x y 5 z 3
Interchange the first and the second equations of the system:
x y z 0
3 x 2 y z 5
4 x y 5 z 3
Subtract from the second equation the first equation multiplied on
3; also subtract from the third equation the first equation
multiplied on 4. We obtain:
x y z 0
y 4z 5
5 y 9z 3

10.

Further subtract from the third equation the second equation
multiplied on 5:
x y z 0
y 4z 5
11z 22
Multiply the second equation on (-2), and the third – divide on
(-11):
x y z 0
y 4 z 5
z 2
The system of equations has a triangular form, and consequently
it has a unique decision. From the last equation we have z = 2;
substituting this value in the second equation, we receive у = 3
and, at last from the first equation we find х = -1.

11.

• Maricopa's Success scholarship fund receives
a gift of $85000. The money is invested in
stocks, bonds, and CDs. CDs pay 3.25%
interest, bonds pay 4.1% interest, and stocks
pay 7.7% interest. Maricopa Success invests
$15000 more in bonds than in CDs. If the
annual income from the investments is
$3992.5. How much was invested in each
account?