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 ? 

Let A = (a$$_{i,j}$$​ ) be a real symmetric 3 × 3 matrix. Consider the quadratic form

$$Q(x_1, x_2, x_3) = \mathbf{x}^T A \mathbf{x}, \quad \text{where } \mathbf{x} = (x_1, x_2, x_3)^T$$​.

Which of the following is true?

Suppose A and B are similar real matrices, that is, there exists an invertible matrix S such that $$A = S B S^{-1}.$$​
Which of the following need not be true?

Let  A be an invertible $$5 \times 5$$​ matrix over a field  F. Suppose that the characteristic polynomials of A and  $$A^{-1}$$​ are the same.
Which of the following is necessarily true?

Consider the linear programming problem:

Maximize z = 3x + 4y

Subject to :

$$\text{Let } M \text{ be a } 5 \times 5 \text{ matrix with real entries such that } \\ \text{Rank}(M) = 3. \\ \text{Consider the linear system } Mx = b. \\ \text{Let the row-reduced echelon form of the augmented matrix } [M \, b] \\ \text{be } R, \text{ and let } R[i, :] \text{ denote the } i\text{-th row of } R. \\ \text{Suppose that the linear system admits a solution. Which of the following statements is necessarily true?}$$​

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

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

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