Let $A$ be a $10\times 10$ real matrix. Assume that the rank of $A$ is 7. Which of the following statements is necessarily true?​

Let A be a $$2 \times2$$​ real matrix with detA=1 and trace A=3.

What is the value of trace of $$A^2?$$​​

Let a , b and c are distinct integers. Let A be a matrix

​$$A = \begin{pmatrix}a^2 & b^2 & c^2 \\a^5 & b^5 & c^5 \\a^{11} & b^{11} & c^{11}\end{pmatrix}$$

Which among the following is the set of all possible ranks of A ? 

Consider the linear programming problem:

Maximize z = 3x + 4y

Subject to :

​$$\text{Let } A \text{ be an } n \times n \text{ matrix of rank 1. Let } \alpha = \det(I + A),\\ \text{ where } I \text{ is the identity matrix and let } \beta = \text{trace}(A).\\ \text{ Which of the following is true?}$$​​

​$$Let \ A\ \text{ be an } n \times n \text{ matrix such that the set of all its nonzero}\\ \text{eigenvalues has exactly r elements.}\\\text{ Which of the following statements is true?}$$​​

​$$\text{If } A = \begin{pmatrix}3 & -2 \\2 & -1\end{pmatrix}, \text{ then } A^{20} \text{ equals:}$$​​

Let A and B be a $$2\times 2$$ matrices. Then which of the following is true ?​

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