TS EAMCET 11 May 2024 Shift 1 Solved Paper

Given the matrices A and B :
A=[

1	2
2	1

],‌‌B=[

x	y
1	2

]
We have the condition:
(A+B)(A−B)=A2−B2
This implies:
AB−BA=0‌‌⇒‌‌AB=BA
Let's compute AB and BA :
‌AB=[

1	2
2	1

][

x	y
1	2

]=[

x+2	y+4
2x+1	2y+2

]
‌BA=[

x	y
1	2

][

1	2
2	1

]=[

x+2y	2x+y
1+4	2+2

]=[

x+2y	2x+y
5	4

]
Equating AB and BA, we have:
[

x+2	y+4
2x+1	2y+2

]=[

x+2y	2x+y
5	4

]
From this, we derive:
‌2x+1=5
‌2y+2=4

Thus:
‌2x=4‌‌⇒‌‌x=2
‌2y=2‌‌⇒‌‌y=1
Given matrix C :
C=[

x	2
1	y

]=[

2	2
1	1

]
The trace of matrix C, denoted as tr(C), is:
tr(C)=x+y=2+1=3