حل السؤال الرابع ..
A = 7 -4
5 -2
det (A - I) = 0 // here I is the identity matrix
7 -4 0
- = 0
5 -2 0
we get
7 - -4 = 0
5 -2 -
(7 - ) (-2 - ) - (-4) * 5 = 0
-14 -7 + 2 + 2 + 20 = 0
2 - 5 + 6 = 0
2 - 3 - 2 + 6 = 0
( - 3) -2 ( - 3) = 0
( - 3) ( - 2) = 0
= 3 , 2
So the Eigenvalues are 3, 2.
Eigen vectors
A - I : 0
7 - -4 0 // here = 3
5 -2 - 0
7 - 3 -4 0
5 -2 - 3 0
4 -4 0
5 -5 0
R1 -> R1/4 and R2 ->R2/5
1 -1 0
1 -1 0
R2->R2 - R1
1 -1 0
0 0 0
rewriting this augmented matrix as linear equation we get
x - y = 0
x = y
we get x = 1 and y = 1
Eigen vector for = 3 is (1,1)
for = 2
7 - -4 0
5 -2 - 0
5 -4 0
5 -4 0
R1->R1/5 and R2 ->R2/5
1 -4/5 0
1 -4/5 0
rewriting this augmented matrix as linear equation we get
x - 4y/5 = 0
5x - 4 y = 0
5x = 4y
x/y = 4/5
(4,5)
العرض العادي
