JEE MAIN - Mathematics (2021 - 26th February Morning Shift - No. 3)
Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is :
6
4
1
12
Explanation
Let $$A = \left[ {\matrix{
a & b \cr
b & c \cr
} } \right]$$
$${A^2} = \left[ {\matrix{ a & b \cr b & c \cr } } \right]\left[ {\matrix{ a & b \cr b & c \cr } } \right] = \left[ {\matrix{ {{a^2} + {b^2}} & {ab + bc} \cr {ab + bc} & {{c^2} + {b^2}} \cr } } \right]$$
$$ = {a^2} + 2{b^2} + {c^2} = 1$$
$$a = 1,b = 0,c = 0$$
$$a = 0,b = 0,c = 1$$
$$a = - 1,b = 0,c = 0$$
$$c = - 1,b = 0,a = 0$$
$${A^2} = \left[ {\matrix{ a & b \cr b & c \cr } } \right]\left[ {\matrix{ a & b \cr b & c \cr } } \right] = \left[ {\matrix{ {{a^2} + {b^2}} & {ab + bc} \cr {ab + bc} & {{c^2} + {b^2}} \cr } } \right]$$
$$ = {a^2} + 2{b^2} + {c^2} = 1$$
$$a = 1,b = 0,c = 0$$
$$a = 0,b = 0,c = 1$$
$$a = - 1,b = 0,c = 0$$
$$c = - 1,b = 0,a = 0$$
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