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Byju's Answer
Standard XII
Mathematics
Scalar Matrix
Let A be a ...
Question
Let
A
be a square matrix of order
n
×
n
. A constant
λ
is said to be characteristic root of
A
if there exists a
n
×
1
matrix
X
such that
A
X
=
λ
X
If
λ
is a characteristic root of
A
, then :
A
A
−
λ
I
=
0
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B
A
−
λ
I
is singular
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C
A
−
λ
I
is non-singular
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D
none of these
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Solution
The correct option is
B
A
−
λ
I
is singular
Since
X
≠
0
is such that
(
A
−
λ
I
)
X
=
0
,
|
A
−
λ
I
|
=
0
⇔
A
−
λ
I
is singular.
If
A
−
λ
I
is non-singular the then equation
(
A
−
λ
I
)
X
=
0
⇒
X
=
0
If
λ
=
0
, we get
|
A
|
=
0
⇒
A
is singular.
We have
A
2
X
=
A
(
A
X
)
=
A
(
λ
X
)
=
λ
(
A
X
)
=
λ
2
X
A
3
X
=
A
(
A
2
X
)
=
A
(
λ
2
X
)
=
λ
2
(
A
X
)
=
λ
2
(
λ
X
)
=
λ
3
X
Continuing in this way, we obtain
A
"
X
=
λ
′′
X
∀
n
∈
N
Also,
|
P
−
1
A
P
−
λ
I
|
=
|
P
−
1
(
A
−
λ
I
)
P
|
=
|
P
−
1
|
|
A
−
λ
I
|
|
P
|
=
|
A
−
λ
I
|
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0
Similar questions
Q.
Let
A
be a square matrix of order
n
×
n
. A constant
λ
is said to be characteristic root of
A
if there exists a
n
×
1
matrix
X
such that
A
X
=
λ
X
If
0
is a characteristic root of
A
, then :
Q.
Let
A
be a square matrix of order
n
×
n
. A constant
λ
is said to be characteristic root of
A
if there exists a
n
×
1
matrix
X
such that
A
X
=
λ
X
If
λ
is a characteristic root of
A
and
n
∈
N
, then
λ
n
is a characteristic root of
Q.
Let
A
be a square matrix of order
n
×
n
. A constant
λ
is said to be characteristic root of
A
if there exists a
n
×
1
matrix
X
such that
A
X
=
λ
X
Let
P
be a non-singular matrix, then which of the following matrices have the same characteristic roots.
Q.
Let
A
be a square matrix of order
n
×
n
and let
P
be a non-singular matrix, then which of the following matrices have the same characteristic roots.
Q.
Let A be a square matrix of order 2 or 3 and I will be the identity matrix of the same order. Then the matrix A -
λ
I is called the characteristic matrix of the matrix A, where
λ
is some complex number. The determinant of the characteristic matrix is called characteristic determinant of matrix A which will, of course, be a polynomial of degree 3 in
λ
. The equation det (A -
λ
I)
=
0 is called the characteristic equation of the matrix A and its roots (the values of
λ
) are called characteristic roots or eigenvalues. It is also known that every square matrix has its characteristic equation.
The eigenvalues of the matrix
A
=
⎡
⎢
⎣
2
1
1
2
3
4
−
1
−
1
−
2
⎤
⎥
⎦
are
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