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

Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementary row operation ?

A
$$\left[\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right]$$
B
$$\left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right]$$
C
$$\left[\begin{array}{rr}-1 & 2 \\ -2 & 7\end{array}\right]$$
D
$$\left[\begin{array}{ll}-1 & 2 \\ -1 & 3\end{array}\right]$$
2
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

If the system of equations

\begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned}

has infinitely many solutions, then $$\alpha+\beta$$ is equal to

A
8
B
36
C
44
D
48
3
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
4
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
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