Properties III

Q. The matrix A is defined as $A=\left[\begin{array}{ccc}1& 2& -3\\ 0& 3& 2\\ 0& 0& -2\end{array}\right]$ the determinant of matrix $B=3A^3+5A^2-6A+2I$ is

1. 110
2. 1100
3. 4400
4. 440

Q. -8 and 5 are the eigen value of a non-singular matrix A, then the eigen values of adj A are

1. 16, -10
2. -5, 8
3. 5, -8
4. -16, 10

Q. Consider the matrix A = $\left[\begin{array}{ccc}2& 1& 1\\ 2& 3& 4\\ -1& -1& -2\end{array}\right]$ whose eigen values are 1, -1 and 3. The trace $(A^3-3A^2)$ is

1. -6

Q. The matrix $M=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& 6\\ -1& -2& 0\end{array}\right]$ has eigen values 3, -3, 5. An eigen vector corresponding to the eigen value 5 is $[12-1{]}^T$. One of the eigen vector of the matrix $M^3$ is

1. ${\left[\begin{array}{ccc}1& 8& -1\end{array}\right]}^T$
2. ${\left[\begin{array}{ccc}1& 2& -1\end{array}\right]}^T$
3. ${\left[\begin{array}{ccc}1& \sqrt[3]{32}& -1\end{array}\right]}^T$
4. ${\left[\begin{array}{ccc}1& 1& -1\end{array}\right]}^T$

Q.

If $f\left(x\right)=\frac{x}{1+x}$ and $g\left(x\right)=f\left[f\left(x\right)\right]$ then $g\left(x\right)$is equal to

1. $\frac{1}{{\left(2x+3\right)}^2}$

2. $\frac{1}{{\left(x+1\right)}^2}$

3. $\frac{1}{x^2}$

4. $\frac{1}{{\left(2x+1\right)}^2}$

Q. If a square matrix of order 100 has exactly 15 distinct eigen values, then the degree of the minimal polynomial is

1. At least 15
2. At most 15
3. Always 15
4. Exactly 100