1
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a 3 $$\times$$ 3 matrix A such that $$A^2=3A+\alpha I$$. If $$A^4=21A+\beta I$$, then

A
$$\alpha=1$$
B
$$\alpha=4$$
C
$$\beta=8$$
D
$$\beta=-8$$
2
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Consider the following system of equations

$$\alpha x+2y+z=1$$

$$2\alpha x+3y+z=1$$

$$3x+\alpha y+2z=\beta$$

for some $$\alpha,\beta\in \mathbb{R}$$. Then which of the following is NOT correct.

A
It has a solution for all $$\alpha\ne-1$$ and $$\beta=2$$
B
It has no solution if $$\alpha=-1$$ and $$\beta\ne2$$
C
It has no solution for $$\alpha=-1$$ and for all $$\beta \in \mathbb{R}$$
D
It has no solution for $$\alpha=3$$ and for all $$\beta\ne2$$
3
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1 Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) A$$^{13}$$ B$$^{26}$$ $$-$$ B$$^{26}$$ A$$^{13}$$ is symmetric

(S2) A$$^{26}$$ C$$^{13}$$ $$-$$ C$$^{13}$$ A$$^{26}$$ is symmetric

Then,

A
Only S2 is true
B
Only S1 is true
C
Both S1 and S2 are false
D
Both S1 and S2 are true
4
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1 Let $$A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - i} \cr 0 & 1 \cr } } \right]$$, where $$i = \sqrt { - 1}$$. If $$\mathrm{M=A^T B A}$$, then the inverse of the matrix $$\mathrm{AM^{2023}A^T}$$ is

A
$$\left[ {\matrix{ 1 & { - 2023i} \cr 0 & 1 \cr } } \right]$$
B
$$\left[ {\matrix{ 1 & 0 \cr {2023i} & 1 \cr } } \right]$$
C
$$\left[ {\matrix{ 1 & {2023i} \cr 0 & 1 \cr } } \right]$$
D
$$\left[ {\matrix{ 1 & 0 \cr { - 2023i} & 1 \cr } } \right]$$
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