Properties I

Q. The three characteristic roots of the following matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 0& 2\end{array}\right]$ are

1. $1,2,3$
2. $1,2,2$
3. $1,0,0$
4. $0,2,3$

Q. Eigen values of the matrix $A=\left[\begin{array}{ccc}3& 1& 4\\ 0& 2& 6\\ 0& 0& 5\end{array}\right]$ are

1. 2, 3, 5
2. 1, 2, 3
3. 1, 2, 5
4. 3, 5, 8

Q. The trace and determinant of a 2 x 2 matrix are known to be -2 and -35 respectively. Its eigen values are

1. $-30and-5$
2. $-37and-1$
3. $-7and5$
4. $17.5and-2$

Q. Eigen values of the Matrix $\left[\begin{array}{ccc}3& -1& -1\\ -1& 3& -1\\ -1& -1& 3\end{array}\right]$ are

1. 1, 1, 1
2. 1, 1, 2
3. 1, 4, 4
4. 1, 2, 4

Q. Consider the following matrix $A=\left[\begin{array}{cc}2& 3\\ x& y\end{array}\right]$. If the eigen values of A are 4 an 8, then

1. $x=4,y=10$
2. $x=5,y=8$
3. $x=-3,y=9$
4. $x=-4,y=10$

Q.

The eigen values of $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ are

1. $0,0,0$
2. $0,0,1$
3. $0,0,3$
4. $1,1,1$

Q. The sum of the eigen values of the matrix given below is $\left[\begin{array}{ccc}1& 2& 3\\ 1& 5& 1\\ 3& 1& 1\end{array}\right]$

1. $5$
2. $7$
3. $9$
4. $18$

Q.

Three non-zero real numbers form A.P. and the squares of these numbers taken in the same order form a G.P. Then the number of all possible common ratios of the G.P. is

1. $1$

2. $2$

3. $3$

4. None of these

Q. The matrix $\left[\begin{array}{ccc}1& 2& 4\\ 3& 0& 6\\ 1& 1& p\end{array}\right]$ has one eigen value equal to 3. The sum of the other two eigen values is

1. $p$
2. $p-1$
3. $p-2$
4. $p-3$

Q. The product of eigen values of the matrix is P is

P = $\left[\begin{array}{ccc}2& 0& 1\\ 4& -3& 3\\ 0& 2& -1\end{array}\right]$

1. -6
2. 2
3. 6
4. -2

Q. The value of x for which the matrix $A=\left[\begin{array}{ccc}3& 2& 4\\ 9& 7& 13\\ -6& -4& -9+x\end{array}\right]$ has zero as an eigen value is ______ .

1. 1

Q. Let M be a real 4 x 4 matrix. Consider the following statements:

S1 : M has 4 linearly independent eigen vectors.

S2 : M has 4 distinct eigen values.

S3 : M is non-singular (invertible).

Which one among the following is TRUE?

1. S1 implies S2
2. S1 implies S3
3. S2 implies S1
4. S3 implies S2

Q. If the entries in each column of a square matrix M add upto 1, then an eigen value of M is

1. 4
2. 3
3. 2
4. 1

Q. The eigen values of the matrix $M$ given below are $15,3$ and $0.M=\left[\begin{array}{ccc}8& -6& 2\\ -6& 7& -4\\ 2& -4& 3\end{array}\right]$
The value of the determinant of the matrix is

1. $20$
2. $10$
3. $0$
4. $-10$

Q. Consider a non-singular 2 x 2 square matrix A. If trance (A) = 4 and trace $(A{)}^2$ = 5, the determinant of the matrix A is (upto 1 decimal place).

Q. Consider the matrix $\left[\begin{array}{cc}5& -1\\ 4& 1\end{array}\right]$ which one of the following statements is TRUE for the eigen values and eigen vectors of the matrix?

1. Eigen value 3 has a multiplicity of 2 and only one independent eigen vector exists
2. Eigen value 3 has a multiplicity of 2 and two independent eigen vectors exist
3. Eigen value 3 has a multiplicity of 2 and no independent eigent vector exists
4. Eigen values are 3 and -3 and two independent eighen vectors exist.

Q. $A=\left[\begin{array}{cccc}2& 0& 0& -1\\ 0& 1& 0& 0\\ 0& 0& 3& 0\\ -1& 0& 0& 4\end{array}\right]$. The sum of the eigen values of the matrix A is

1. $10$
2. $-10$
3. $24$
4. $22$

Q. Let A be n x n real valued square symmetric matrix of rank 2 with ${\sum }_{i=1}^n{\sum }_{j=1}^n{A}_{ij}^2$ = 50. Consider the following statements.

(I) One eigen value must be in [-5, 5]
(II) The eigen value with the largest magnitude must be strictly greater than 5.

Which of the above statementss about eigen values of A is/are necessary CORRECT?

1. Both (I) and (II)
2. (I) Only
3. (II) only
4. Neither (I) nor (II)

Q. Eigen values of the matrix $A=\left[\begin{array}{ccc}3& 1& 4\\ 0& 2& 6\\ 0& 0& 5\end{array}\right]$ are

1. 2, 3, 5
2. 1, 2, 3
3. 1, 2, 5
4. 3, 5, 8