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MATRICES
BY ALFIA MAGFIRONAD100102004
CIVIL ENGINEERING DEPARTEMENTENGINEERING FACULTY
MUHAMMADIYAH UNIVERSITY OF SURAKARTA
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.
333231
232221
131211
aaa
aaa
aaa
A
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
eg.
3332
232211 aa
aam
Therefore the minor of a12 is:
And the minor for a13 is:
3331
232112 aa
aam
3231
222113 aa
aam
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:
ijji
ij mC )1(
When the sum of a row number i and column j is even, cij = mij and when i+j is odd, cij =-mij
131331
13
121221
12
111111
11
)1()3,1(
)1()2,1(
)1()1,1(
mmjic
mmjic
mmjic
333231
232221
131211
333231
232221
131211
MMM
MMM
MMM
CCC
CCC
CCC
The Formula :
DETERMINANTS CONTINUED
The determinant of an n x n matrix A can now be defined as
nncacacaAA 1112121111det
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)
Therefore the 2 x 2 matrix :
2221
1211
aa
aaA
Has cofactors :
22221111 aamc
And:
21211212 aamc
For a 3 x 3 matrix:
333231
232221
131211
aaa
aaa
aaa
A
The cofactors of the first row are:
312232213231
222113
312333213331
232112
322333223332
232211
)(
aaaaaa
aac
aaaaaa
aac
aaaaaa
aac
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.
A =
first diagonal of matrix
second diagonal of matrix
Adj A =
PROBLEM
Find Adjoint of matrix
We can use the formula of The adjoint matrix for 2 x 2 square matrix.So,
Adj
The adjoint matrix for 3 x 3 square matrix
OR
To determine the adjoint matrix for 3 x 3 square matrix is used cofactor matrix in each elements in the square of matrix.
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 (-)
PROBLEM
Find Adjoint of matrix
Solution :
OR
Adj
or
Adj
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. 𝑨−𝟏 = 𝒂𝒅𝒋 𝑨ȁ𝑨ȁ the inverse of matrix 𝑨= (𝒂𝒊𝒋)
b. 𝑨𝑨−𝟏 = 𝑨−𝟏𝑨= 𝑰 (𝒊𝒅𝒆𝒏𝒕𝒊𝒕𝒚) For any nonsingular matrix A c. ሺ𝒂𝒅𝒋𝑨ሻ𝑨= 𝑨ሺ𝒂𝒅𝒋𝑨ሻ= ȁ𝑨ȁ𝑰 For any square matrix A
d. ห𝑨−𝟏ห= 𝟏ȁ𝑨ȁ If A is nonsingular
e. 𝑨𝑿= 𝑩, 𝑿= 𝑨−𝟏𝑩 If A is an m x n nonsingular matrix, 𝑿𝑨= 𝑩, 𝑿= 𝑩𝑨−𝟏 If B is an n x m matrix, and there
exist matrix X f. ሺ𝑨𝑩ሻ−𝟏 = 𝑩−𝟏𝑨−𝟏 For any two nonsingular matrices A and B
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
EXAMPLE
1.03.0
2.04.0
13
24
10
11A
43
21A =
To check AA-1 = A-1 A = I
IAA
IAA
10
01
43
21
1.03.0
2.04.0
10
01
1.03.0
2.04.0
43
21
1
1
Example 2
121
012
113
A
|A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2
),1(
),1(
),1(
31
21
11
c
c
c
The determinant of A is
The elements of the cofactor matrix are
),2(
),4(
),2(
32
22
12
c
c
c
),5(
),7(
),3(
33
23
13
c
c
c
521
741
321
C
The cofactor matrix is therefore
so
573
242
111TCadjA
and
5.25.35.1
0.10.20.1
5.05.05.0
573
242
111
2
11
A
adjAA