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Byju's Answer
Standard XII
Mathematics
Matrix Definition and Representation
If A = [ [ ...
Question
If
A
=
⎡
⎢
⎣
0
4
3
1
−
3
−
3
−
1
4
4
⎤
⎥
⎦
then prove that
A
2
=
1
.
Hence show that
A
−
1
=
A
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Solution
A
=
⎡
⎢
⎣
0
4
3
1
−
3
−
3
−
1
4
4
⎤
⎥
⎦
A
2
=
⎡
⎢
⎣
0
4
3
1
−
3
−
3
−
1
4
4
⎤
⎥
⎦
⎡
⎢
⎣
0
4
3
1
−
3
−
3
−
1
4
4
⎤
⎥
⎦
A
2
=
⎡
⎢
⎣
1
0
0
0
1
0
0
0
1
⎤
⎥
⎦
=
I
A
=
A
−
1
Hence, proved.
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Similar questions
Q.
If
A
=
∣
∣ ∣
∣
1
2
2
2
1
2
2
2
1
∣
∣ ∣
∣
, then prove that
A
2
−
4
A
−
5
I
3
=
0
and hence obtain
A
−
1
.
Q.
If
A
=
⎡
⎢
⎣
1
2
2
2
1
2
2
2
1
⎤
⎥
⎦
, then show that
A
2
−
4
A
−
5
I
=
0
, and hence find
A
−
1
.
Q.
Prove that :
A
2
−
6
A
−
17
I
2
=
0
, where
A
=
[
2
−
3
3
4
]
and hence if det
(
A
−
1
)
=
1
m
.Find
m
Q.
a) Prove that
a
∫
a
f
(
x
)
d
x
=
⎧
⎪ ⎪ ⎪
⎨
⎪ ⎪ ⎪
⎩
2
a
∫
0
f
(
x
)
d
x
if
f
(
x
)
is an even function
0
if
f
(
x
)
is an odd function
and hence evaluate
1
∫
−
1
sin
5
x
cos
4
x
d
x
.
b) Prove that
∣
∣ ∣ ∣
∣
a
2
+
1
a
b
a
c
a
b
b
2
+
1
b
c
c
a
c
b
c
2
+
1
∣
∣ ∣ ∣
∣
=
1
+
a
2
+
b
2
+
c
2
.
Q.
If
A
=
[
3
1
−
1
2
]
show that
A
2
−
5
A
+
7
I
=
O
. Hence , Find
A
−
1
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