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The eigen values of a (2 × 2) matrix X are -2 and -3. The eigen values of the matrix (X + $$I$$) (X + 5$$I$$) are
A
-3, -4
B
-1, -2
C
-1, -3
D
-2, -4
Correct Answer:
-3, -4
Let the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x
1
and x
2
respectively. Then the eigen values and eigen vectors of the matrix A
2
- 3A + 4$$I$$ would, respectively, be
A
2, 14; x<sub>1</sub>, x<sub>2</sub>
B
2, 14; x<sub>1</sub> + x<sub>2</sub>, x<sub>1</sub> - x<sub>2</sub>
C
2, 0; x<sub>1</sub>, x<sub>2</sub>
D
2, 0; x<sub>1</sub> + x<sub>2</sub>, x<sub>1</sub> - x<sub>2</sub>
Consider the system of equations A
(n × n)
X
(n × 1)
= λ
(n × 1)
where, λ is a scalar. Let (λ
i
, x
i
) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a(n × n) unit matrix. Which one of the following statement is NOT correct?
A
For a homogeneous n × n system of linear equations, (A - λ$$I$$)x = 0 having a nontrivial solution, the rank of (A - λ$$I$$) is less than n
B
For matrix A<sup>m</sup>, m being a positive integer, (λ<sub>i</sub><sup>m</sup>, x<sub>i</sub><sup>m</sup>) will be the eigen-pair for all i
C
If A<sup>T</sup> = A<sup>-1</sup>, then |λ<sub>i</sub>| = 1 for all i
D
If A<sup>T</sup> = A, then λ<sub>i</sub> is real for all i
A 3 × 3 matrix has elements such that its trace is 11 and its determinant is 36. The eigen values of the matrix are all known to be positive integers. The largest eigen value of the matrix is
A
18
B
12
C
9
D
6
Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.
A
a, b, c
B
-a, -b, -c
C
a-b, b-a, c-a
D
a-1, b-1, c-1
The Eigen values of a 3×3 matrix are λ1, λ2, λ3 then the Eigen values of a matrix A3 are __________
A
λ1, λ2, λ3
B
\( \frac{1}{λ_1}, \frac{1}{λ_2}, \frac{1}{λ_3}\)
C
\(λ_1^3, λ_2^3, λ_3^3\)
D
1, 1, 1
A real traceless 4 × 4 matrix has to eigen values -1 and +1. The other eigen values are
A
zero and +2
B
-1 and +1
C
zero and +1
D
+1 and +1
M is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M
2
are
A
-2 and -3
B
2 and 3
C
4 and 9
D
16 and 81
Let the matrix A be the idempotent matrix then the Eigen values of the idempotent matrix are ________
A
0, 1
B
0
C
1
D
-1
The determinant of a 3 × 3 real symmetric matrix is 36. If two of its eigen values are 2 and 3 then the third eigen value is
A
4
B
6
C
8
D
9
The trace of a 3 × 3 matrix is 2. Two of its eigen values are 1 and 2. The third eigen value is
A
-1
B
0
C
1
D
2