all about matriks

1. MATRICES
BY ALFIA MAGFIRONA
D100102004
CIVIL ENGINEERING DEPARTEMENT
ENGINEERING FACULTY
MUHAMMADIYAH UNIVERSITY OF SURAKARTA

2. MATRICES - OPERATIONS
MINORS
If A is an n x n matrix and one row and one column are deleted, the resulting
matrix is an (n-1) x (n-1) submatrix of A.
The determinant of such a submatrix is called a minor of A and is designated
by mij , where i and j correspond to the deleted
row and column, respectively.
mij is the minor of the element aij in A.

3. eg.
a11 a12 a13
A a21 a22 a23
a31 a32 a33
Each element in A has a minor
Delete first row and column from A .
The determinant of the remaining 2 x 2 submatrix is the minor of a11
a22 a23
m11
a32 a33

4. Therefore the minor of a12 is:
a21 a23
m12
a31 a33
And the minor for a13 is:
a21 a22
m13
a31 a32

5. E. COFACTOR OF MATRIX
If A is a square matrix, then the minor of its entry aij, also known as the
i,j, or (i,j), or (i,j)th minor of A, is denoted by Mij and is defined to be the
determinant of the submatrix obtained by removing from A its i-th row
and j-th column. It follows:
Cij ( 1)i j mij
When the sum of a row number i and column j is even, cij = mij and
when i+j is odd, cij =-mij
c11 (i 1, j 1) ( 1)1 1 m11 m11
c12 (i 1, j 2) ( 1)1 2 m12 m12
1 3
c13 (i 1, j 3) ( 1) m13 m13

6. The Formula :
C11 C12 C13 M 11 M 12 M 13
C21 C22 C23 M 21 M 22 M 23
C31 C32 C33 M 31 M 32 M 33

7. DETERMINANTS CONTINUED
The determinant of an n x n matrix A can now be defined as
A det A a11c11 a12c12  a1nc1n
The determinant of A is therefore the sum of the products of the
elements of the first row of A and their corresponding cofactors.
(It is possible to define |A| in terms of any other row or column but for
simplicity, the first row only is used)

8. Therefore the 2 x 2 matrix :
a11 a12
A
a21 a22
Has cofactors :
c11 m11 a22 a22
And:
c12 m12 a21 a21

9. For a 3 x 3 matrix:
a11 a12 a13
A a21 a22 a23
a31 a32 a33
The cofactors of the first row are:
a22 a23
c11 a22 a33 a23a32
a32 a33
a21 a23
c12 (a21a33 a23a31 )
a31 a33
a21 a22
c13 a21a32 a22 a31
a31 a32

10. F. ADJOINT OF MATRIX
 The adjoint matrix for 2 x 2 square matrix
A= , so Adjoint of matrix A is
Elements in the first diagonal of matrix is
exchanged, and the second diagonal of matrix is
just changed mark.

11. A= second diagonal of
matrix
first diagonal of
matrix
Adj A =

12. PROBLEM
Find Adjoint of matrix
We can use the formula of The adjoint matrix for 2 x 2
square matrix.
So,
Adj

13.  The adjoint matrix for 3 x 3 square matrix
OR

14. To determine the adjoint matrix for 3 x 3 square
matrix is used cofactor matrix in each elements in the
square of matrix.

15. It uses cofactor of matrix A1.1 to fill in
fisrt rows of A and for the others we
must use others cofactor.
Don’t forget to obseve the
mark : (+) or (-)

16. PROBLEM
Find Adjoint of matrix
Solution :
OR

17. Adj
or
Adj

18. G. INVERSE OF MATRIX
It is easy to show that the inverse of matrix is uniqe and the
inverse of the inverse of A is A-1 but there is also many
properties inverse matix; that is,
������������������ ������
a. ������−������ = the inverse of matrix ������ = (������������������ )
������
b. ������������−������ = ������ ������ = ������ (������������������������������������������������)
−������
For any nonsingular matrix A
c. ������������������������ ������ = ������ ������������������������ = ������ ������ For any square matrix A
������
d. ������−������ = If A is nonsingular
������
e. ������������ = ������, ������ = ������−������ ������ If A is an m x n nonsingular matrix,
������������ = ������, ������ = ������������−������ If B is an n x m matrix, and there
exist matrix X
f. ������������ −������ = ������−������ ������−������ For any two nonsingular matrices A and B

19.  A square matrix that has an inverse is called
a nonsingular matrix
 A matrix that does not have an inverse is
called a singular matrix
 Square matrices have inverses except when
the determinant is zero
 When the determinant of a matrix is zero the
matrix is singular

20. EXAMPLE
1 2
A=
3 4
1 1 4 2 0.4 0.2
A
10 3 1 0.3 0.1
To check AA-1 = A-1 A = I
1 1 2 0.4 0.2 1 0
AA I
3 4 0.3 0.1 0 1
1 0.4 0.2 1 2 1 0
A A I
0.3 0.1 3 4 0 1

21. Example 2
3 1 1
A 2 1 0
1 2 1
The determinant of A is
|A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2
The elements of the cofactor matrix are
c11 ( 1), c12 ( 2), c13 (3),
c21 ( 1), c22 ( 4), c23 (7),
c31 ( 1), c32 ( 2), c33 (5),

22. The cofactor matrix is therefore
1 2 3
C 1 4 7
1 2 5
so
1 1 1
adjA C T 2 4 2
3 7 5
and
1 1 1 0.5 0.5 0.5
1 adjA 1
A 2 4 2 1.0 2.0 1.0
A 2
3 7 5 1.5 3.5 2.5