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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
Correct Answer:
9
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>
Let A be an m × n matrix and Ban n × m matrix. It is given that determinant ($$I$$
m
+ AB) = determinant ($$I$$
n
+ BA), where $$I$$
k
is the k × k identity matrix. Using the above property, the determinant of the matrix given below is
\[\left[ {\begin{array}{*{20}{c}} 2&1&1&1 \\ 1&2&1&1 \\ 1&1&2&1 \\ 1&1&1&2 \end{array}} \right
A
\
B
<p><span>A.</span> 2
C
</span> 5
Consider the following statements and select which of the following statement are true. Statement 1: The maximum sum rectangle can be 1X1 matrix containing the largest element If the matrix size is 1X1 Statement 2: The maximum sum rectangle can be 1X1 matrix containing the largest element If all the elements are zero Statement 3: The maximum sum rectangle can be 1X1 matrix containing the largest element If all the elements are negative
A
Only statement 1 is correct
B
Only statement 1 and Statement 2 are correct
C
Only statement 1 and Statement 3 are correct
D
Statement 1, Statement 2 and Statement 3 are correct
The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. It eigen values are
A
-30 and -5
B
-37 and -1
C
-7 and 5
D
17.5 and -2
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
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
Let Z be the set of real integers and R the set of real numbers. The sampling process may be viewed as partitioning the x-y plane into a grid, with the central coordinates of each grid being from the Cartesian product Z2, that is a set of all ordered pairs (zi, zj), with zi and zj being integers from Z. Then, f(x, y) is a digital image if (x, y) are integers from Z2 and f is a function that assigns a gray-level value (that is, a real number from the set R) to each distinct coordinate pair (x, y). What happens to the digital image if the gray levels also are integers?
A
The Digital image then becomes a 2-D function whose coordinates and amplitude values are integers
B
The Digital image then becomes a 1-D function whose coordinates and amplitude values are integers
C
The gray level can never be integer
D
None of the mentioned
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
The average of 4 positive integers is 59. The highest integer is 83 and the lowest integer is 29. The difference between the remaining two integers is 28. Which of the following integers is higher of the remaining two integers ?
A
39
B
48
C
76
D
Cannot be determined
E
None of these