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M is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M<sup>2</sup> are
A
-2 and -3
B
2 and 3
C
4 and 9
D
16 and 81
Correct Answer:
16 and 81
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>
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
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
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
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
Find out which of the answer figures 1, 2, 3 and 4 completes the figure matrix.
A
1
B
2
C
3
D
4
Find out which of the answer figures 1, 2, 3 and 4 completes the figure matrix.
A
1
B
2
C
3
D
4
Find out which of the answer figures 1, 2, 3 and 4 completes the figure matrix.
A
1
B
2
C
3
D
4
