1
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then

A
the system of linear equations $$A X=0$$ has a unique solution
B
the system of linear equations $$A X=0$$ has infinitely many solutions
C
B is an invertible matrix
D
$$\operatorname{adj}(\mathrm{A})$$ is an invertible matrix
2
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

A
$$\mathrm{A}^{4}-\mathrm{B}^{4}$$ is a smmetric matrix
B
$$\mathrm{AB}-\mathrm{BA}$$ is a symmetric matrix
C
$$\mathrm{B}^{5}-\mathrm{A}^{5}$$ is a skew-symmetric matrix
D
$$\mathrm{AB}+\mathrm{BA}$$ is a skew-symmetric matrix
3
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix $$B_{0}=A^{49}+2 A^{98}$$. If $$B_{n}=A d j\left(B_{n-1}\right)$$ for all $$n \geq 1$$, then $$\operatorname{det}\left(B_{4}\right)$$ is equal to :

A
$$3^{28}$$
B
$$3^{30}$$
C
$$3^{32}$$
D
$$3^{36}$$
4
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1

Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$.

If $$\mathrm{A}^{2}+\gamma \mathrm{A}+18 \mathrm{I}=\mathrm{O}$$, then $$\operatorname{det}(\mathrm{A})$$ is equal to _____________.

A
$$-$$18
B
18
C
$$-$$50
D
50
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