MATRICES BY ALFIA MAGFIRONA D100102004 CIVIL ENGINEERING DEPARTEMENT ENGINEERING FACULTYMUHAMMADIYAH UNIVERSITY OF SURAKARTA
MATRICES - OPERATIONS MINORSIf A is an n x n matrix and one row and one column are deleted, the resultingmatrix is an (n-1) x (n-1) submatrix of A.The determinant of such a submatrix is called a minor of A and is designatedby mij , where i and j correspond to the deletedrow and column, respectively.mij is the minor of the element aij in A.
eg. a11 a12 a13 A a21 a22 a23 a31 a32 a33Each element in A has a minorDelete 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
Therefore the minor of a12 is: a21 a23 m12 a31 a33 And the minor for a13 is: a21 a22 m13 a31 a32
E. COFACTOR OF MATRIXIf A is a square matrix, then the minor of its entry aij, also known as thei,j, or (i,j), or (i,j)th minor of A, is denoted by Mij and is defined to be thedeterminant of the submatrix obtained by removing from A its i-th rowand j-th column. It follows: Cij ( 1)i j mijWhen the sum of a row number i and column j is even, cij = mij andwhen 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
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
DETERMINANTS CONTINUEDThe determinant of an n x n matrix A can now be defined as A det A a11c11 a12c12 a1nc1nThe determinant of A is therefore the sum of the products of theelements 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 forsimplicity, the first row only is used)
Therefore the 2 x 2 matrix : a11 a12 A a21 a22Has cofactors : c11 m11 a22 a22 And: c12 m12 a21 a21
For a 3 x 3 matrix: a11 a12 a13 A a21 a22 a23 a31 a32 a33The 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
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= second diagonal of matrix first diagonal of matrix Adj A =
PROBLEMFind Adjoint of matrixWe can use the formula of The adjoint matrix for 2 x 2square matrix.So, Adj
The adjoint matrix for 3 x 3 square matrix OR
To determine the adjoint matrix for 3 x 3 squarematrix is used cofactor matrix in each elements in thesquare of matrix.
It uses cofactor of matrix A1.1 to fill infisrt rows of A and for the others wemust use others cofactor. Don’t forget to obseve the mark : (+) or (-)
G. INVERSE OF MATRIXIt is easy to show that the inverse of matrix is uniqe and theinverse of the inverse of A is A-1 but there is also manyproperties inverse matix; that is, a. − = the inverse of matrix = ( ) b. − = = () − For any nonsingular matrix Ac. = = 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 Xf. − = − − 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 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
Example 2 3 1 1 A 2 1 0 1 2 1The determinant of A is |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2The elements of the cofactor matrix are c11 ( 1), c12 ( 2), c13 (3), c21 ( 1), c22 ( 4), c23 (7), c31 ( 1), c32 ( 2), c33 (5),
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
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