5.matrix Algebra
5.matrix Algebra
5.matrix Algebra
23
THIRD ORDER DETERMINAT
The determinant of a 3 x 3 matrix can be calculated as follows,
(
(
(
=
33 32 31
23 22 21
13 12 11
a a a
a a a
a a a
A
Each element in a square matrix has its own minor. The minor
is the value of the determinant of the matrix that results from
crossing out the row and column of the element under
consideration.
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MINORS & COFACTORS
i. MINOR
A minor of the given matrix is the determinant of any of its square sub-matrix. Thus, a
minor
ij
M is the determinant of the sub matrix formed by deleting the i th row and
j th column of the matrix.
25
Cofactors
Each element in a square matrix has its own cofactor. The
cofactor is the product of the elements place sign and
minor.
26
i. COFACTOR
A cofactor
ij
C
is a minor with a prescribed sign. The rule for the sign of a cofactor
is
ij
C
=
IJ
j i
M
+
) 1 (
If the sum of subscripts (i + j ) is an even number,
ij
C
=
ij
M
. Since -1,
raise to an even power is positive.
If i + j is equal to an odd number
ij
C
= -
ij
M
. Since, -1 raised to an odd
power is negative.
INVERSE OF MATRIX
Inverse of a matrix can be found only for a square matrix.
The inverse of a matrix [A] is denoted by [A]
-1
.
The product of a matrix and its inverse results in an identity
matrix [I].
The identity matrix [I] has one for the diagonal elements
and all off-diagonal elements are zero
27
28
MATRIX EXPRESSION OF SYSTEM OF LINEAR EQUATION
Matrix algebra permits the concise expression of a system of linear equations. Consider the
following example.
This can be expressed in matrix form
29 5 4
45 3 7
2 1
2 1
= +
= +
x x
x x
29
AX = B
(
=
5 4
3 7
A
(
=
2
1
X
x
x
and
(
=
29
45
B
Here
i. A is the coefficient matrix
ii. X is the solution vector
iii. B is the vector of constant terms
iv. A and B will always be column vectors
30
MATRICES SOLVING TWO SIMULTANEOUS EQUATIONS
One of the most important applications of matrices is to the solution of linear simultaneous
equations. Consider the following simultaneous equation
1 5 3
4 2
=
= +
y x
y x
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CRAMERS RULE FOR MATRIX SOLUTION
Cramers rule provides a simplified method of solving a system of linear equations through
the use of determinants. Cramers rules states that,
A
A
x
i
i
=