1
JEE Main 2021 (Online) 16th March Morning Shift
Numerical
+4
-1
Let $$P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \right]$$ and

$$A = \left[ {\matrix{ 2 & 7 & {{\omega ^2}} \cr { - 1} & { - \omega } & 1 \cr 0 & { - \omega } & { - \omega + 1} \cr } } \right]$$ where

$$\omega = {{ - 1 + i\sqrt 3 } \over 2}$$, and I3 be the identity matrix of order 3. If the
determinant of the matrix (P$$-$$1AP$$-$$I3)2 is $$\alpha$$$$\omega$$2, then the value of $$\alpha$$ is equal to ______________.
2
JEE Main 2021 (Online) 16th March Morning Shift
Numerical
+4
-1
The total number of 3 $$\times$$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to _____________.
3
JEE Main 2021 (Online) 26th February Evening Shift
Numerical
+4
-1
If the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]$$ satisfies the equation

$${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & 0 \cr 0 & 0 & 1 \cr } } \right]$$ for some real numbers $$\alpha$$ and $$\beta$$, then $$\beta$$ $$-$$ $$\alpha$$ is equal to ___________.
4
JEE Main 2021 (Online) 25th February Morning Slot
Numerical
+4
-1
If $$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr } } \right]$$ and
$$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]$$, then $$13({a^2} + {b^2})$$ is equal to
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