Page 152 - Numerical Analysis Using MATLAB and Excel

Chapter 4 Matrices and Determinants

• Some useful properties of determinants are:

a. If all elements of one row or one column are zero, the determinant is zero.

b. If all the elements of one row or column are m times the corresponding elements of
another row or column, the determinant is zero.

c. If two rows or two columns of a matrix are identical, the determinant is zero.

• Cramer’s rule states that if a system of equations is defined as

a x + a y + a z = A
12
11
13
a x + a y + a z = B
21
23
22
a x + a y + a z = C
33
32
31
and we let
a 11 a 12 a 13 Aa 11 a 13 a 11 Aa 13 a 11 a 12 A
Δ = a 21 a 22 a 23 D = Ba 21 a 23 D = a 21 Ba 23 D = a 21 a 22 B
1
2
3
a 31 a 32 a 33 Ca 31 a 33 a 31 Ca 33 a 31 a 32 C
the unknowns , , and can be found from the relations
z
xy
D D D
x = ------ 1 y = ------ 2 z = ------ 3
Δ Δ Δ
provided that the determinant Δ (delta) is not zero.

• We can find the unknowns in a system of two or more equations also by the Gaussian elimina-
tion method. With this method, the objective is to eliminate one unknown at a time. This can
be done by multiplying the terms of any of the equations of the system by a number such that
we can add (or subtract) this equation to another equation in the system so that one of the
unknowns will be eliminated. Then, by substitution to another equation with two unknowns,
we can find the second unknown. Subsequently, substitution of the two values found can be
made into an equation with three unknowns from which we can find the value of the third
unknown. This procedure is repeated until all unknowns are found.

• If is an square matrix and α ij is the cofactor of a ij , the adjoint of , denoted as adjA , is
A
n
A
defined as the square matrix below.
n

4−34 Numerical Analysis Using MATLAB® and Excel®, Third Edition

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