Matrices

Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| $\ne$ 0. Consider the following two statements;

(P)If A $\ne$ I₂, then |A| = – 1

(Q)If |A| = 1, then tr(A) = 2

where I₂ denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

• (P) is true and (Q) are false

• Both (P) and (Q) are true

• Both (P) and (Q) are false

• (P) is false and (Q) is true

Let A = {x = (x, y, z)^T : PX = 0 and x² + y² + z² = 1}, where $\left[\begin{array}{ccc}1& 2& 1\\ - 2& 3& - 4\\ 1& 9& - 1\end{array}\right]$ P then the set A

• is a singleton

• contains more than two elements

• contains exactly two elements

• is an empty set.

Let a, b, c $\in$ R be all non-zero satisfy a³ + b³ + c³ = 2.If the matrix A = $\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right]$ satifies A^TA = I, then a value of abc can be :

• $\frac{1}{3}$

• $- \frac{1}{3}$

• 3

• $\frac{2}{3}$

$\mathrm{Let} \left[\begin{array}{cc}\mathrm{x}& 1\\ 1& 0\end{array}\right] \mathrm{be} \mathrm{a} 2 \times 2 \mathrm{matrix} \mathrm{such} \mathrm{that} {\mathrm{A}}^4 = {\left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{2 \times 2}{\mathrm{a}}_{11} = 109, \mathrm{then} \mathrm{find} {\mathrm{a}}_{22}$

• 12

• 4

• - 8

• 10

$\mathrm{If} \left|\begin{array}{ccc}\mathrm{x} - 2& 2\mathrm{x} - 3& 3\mathrm{x} - 4\\ 2\mathrm{x} - 3& 3\mathrm{x} - 4& 4\mathrm{x} - 5\\ 3\mathrm{x} - 5& 5\mathrm{x} - 8& 10\mathrm{x} - 17\end{array}\right| = {\mathrm{Ax}}^3 + {\mathrm{Bx}}^2 + \mathrm{Cx} + \mathrm{D} \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{absolute} \mathrm{value} \mathrm{B} + \mathrm{C} = ?$

$$\begin{gathered} \mathrm{If} \mathrm{A} =\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \mathrm{isin}\left(\mathrm{\theta }\right)\\ \mathrm{isin}\left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right], \left(\mathrm{\theta } = \frac{\mathrm{\pi }}{24}\right) \mathrm{and} {\mathrm{A}}^5 = \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right], \mathrm{where} \mathrm{i} = \sqrt{ - 1}, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \\ \mathrm{not} \mathrm{true} ? \end{gathered}$$

• $0 \le {\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2 \le 1$

• ${\mathrm{a}}^2 - {\mathrm{d}}^2 = 0$

• ${\mathrm{a}}^2 - {\mathrm{b}}^2 = \frac{1}{2}$

• ${\mathrm{a}}^2 - {\mathrm{c}}^2 = 1$

If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non–zero distinct real numbers then $\left|\begin{array}{ccc}\mathrm{x}& \mathrm{a} + \mathrm{y}& \mathrm{x} + \mathrm{a}\\ \mathrm{y}& \mathrm{b} + \mathrm{y}& \mathrm{y} + \mathrm{b}\\ \mathrm{z}& \mathrm{c} + \mathrm{y}& \mathrm{z} + \mathrm{c}\end{array}\right| = ?$

• y(a - b)

• 0

• y(b - a)

• y(a - c)

Let m and M be respectively the minimum and maximum value values of $\left|\begin{array}{ccc}{\cos }^2\left(\mathrm{x}\right)& 1 + {\sin }^2\left(\mathrm{x}\right)& \sin \left(2\mathrm{x}\right)\\ 1 + {\cos }^2\left(\mathrm{x}\right)& {\sin }^2\left(\mathrm{x}\right)& \sin \left(2\mathrm{x}\right)\\ {\cos }^2\left(\mathrm{x}\right)& {\sin }^2\left(\mathrm{x}\right)& 1 + \sin \left(2\mathrm{x}\right)\end{array}\right|$

Then the ordered pair (m, M) = ?

• ( - 3, 3)

• (1, 3)

• ( - 3, - 1)

• ( - 4, - 1)