1
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1
Change Language

Let $$A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$$ where $$i = \sqrt { - 1} $$. Then, the number of elements in the set { n $$\in$$ {1, 2, ......, 100} : An = A } is ____________.

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2
JEE Main 2022 (Online) 27th June Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

The positive value of the determinant of the matrix A, whose

Adj(Adj(A)) = $$\left( {\matrix{ {14} & {28} & { - 14} \cr { - 14} & {14} & {28} \cr {28} & { - 14} & {14} \cr } } \right)$$, is _____________.

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3
JEE Main 2022 (Online) 26th June Evening Shift
Numerical
+4
-1
Change Language

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 ____________.

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

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 ____________.

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