=2/3
, y = 4/5
and z = 5/6
, prove the following:
I. x (y x z ) = ( x y ) x z
II. x (y + z) = ( x y) + ( x z
Answers
Answer:
Step-by-step explanation:
(i) [
4
x
3
5
]=[
y
1
z
5
]
As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get: x=1,y=4, and z=3
(ii) [
x+y
5+z
2
xy
]=[
6
5
2
8
]
As the given matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
x+y=6,xy=8,5+z=5
Now, 5+z=5⇒z=0
We know that:
(x−y)
2
=(x+y)
2
−4xy
⇒(x−y)
2
=36−32=4
⇒x−y=±2
Now, when x−y=2 and x+y=6, we get x=4 and y=2
When x−y=−2 and x+y=6, we get x=2 and y=4
∴x=4,y=2 and z=0orx=2,y=4 and z=0
(iii)
⎣
⎢
⎢
⎡
x+y+z
x+z
y+z
⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎡
9
5
7
⎦
⎥
⎥
⎤
As the two matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:
x+y+z=9 ....(1)
x+z=5 ....(2)
y+z=7 ....(3)
From (1) and (2), we have:
y+5=9
⇒y=4
Then, from (3), we have: 4+z=7
⇒z=3
∴x+z=5
⇒x=2
∴x=2,y=4 and z=3