1
JEE Main 2022 (Online) 26th June Evening Shift
Numerical
+4
-1

Let $$X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$ and $$Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}$$, $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R. If $${Y^{ - 1}} = \left[ {\matrix{ {{1 \over 5}} & {{{ - 2} \over 5}} & {{1 \over 5}} \cr 0 & {{1 \over 5}} & {{{ - 2} \over 5}} \cr 0 & 0 & {{1 \over 5}} \cr } } \right]$$, then ($$\alpha$$ $$-$$ $$\beta$$ + $$\gamma$$)2 is equal to ____________.

2
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

Let $$A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$$ and $$B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)$$. Then the number of elements in the set {(n, m) : n, m $$\in$$ {1, 2, .........., 10} and nAn + mBm = I} is ____________.

3
JEE Main 2022 (Online) 24th June Evening Shift
Numerical
+4
-1

Let $$S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right\}$$ and let $${T_n} = \{ A \in S:{A^{n(n + 1)}} = I\}$$. Then the number of elements in $$\bigcap\limits_{n = 1}^{100} {{T_n}}$$ is ___________.

4
JEE Main 2021 (Online) 31st August Evening Shift
Numerical
+4
-1
The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}$$, where I is 2 $$\times$$ 2 identity matrix, is :