JEE MAIN - Mathematics (2008 - No. 10)
Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal entries of $$a$$. Assume that $${a^2} = I.$$
Statement-1 : If $$A \ne I$$ and $$A \ne - I$$, then det$$(A)=-1$$
Statement- 2 : If $$A \ne I$$ and $$A \ne - I$$, then tr $$(A)$$ $$ \ne 0$$.
Statement-1 : If $$A \ne I$$ and $$A \ne - I$$, then det$$(A)=-1$$
Statement- 2 : If $$A \ne I$$ and $$A \ne - I$$, then tr $$(A)$$ $$ \ne 0$$.
statement - 1 is false, statement -2 is true
statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.
statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.
statement - 1 is true, statement - 2 is false.
Explanation
Let $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ $$\,\,\,$$ then $${A^2} = 1$$
$$ \Rightarrow {a^2} + bc = 1\,\,\,\,ab + bd = 0$$
$$ac + cd = 0\,\,\,\,bc + {d^2} = 1$$
From these four relations,
$${a^2} + bc = bc + {d^2} \Rightarrow {a^2} = {d^2}$$
and $$\,\,b\left( {a + d} \right) = 0 = c\left( {a + d} \right)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow a = - d$$
We can take $$a = 1,b = 0,c = 0,d = - 1$$
as one possible set of values, then
$$A = \left[ {\matrix{ 1 & 0 \cr 0 & { - 1} \cr } } \right]$$
Clearly $$A \ne I\,\,\,$$ and $$\,\,\,\,A \ne - I\,\,$$ and $$\,\,\,A = - 1$$
$$\therefore$$ $$\,\,\,\,\,$$ Statement $$1$$ is true.
Also if $$A \ne I\,\,\,\,\,tr\left( A \right) = 0$$
$$\therefore$$ $$\,\,\,\,\,$$ Statement $$2$$ is false.
$$ \Rightarrow {a^2} + bc = 1\,\,\,\,ab + bd = 0$$
$$ac + cd = 0\,\,\,\,bc + {d^2} = 1$$
From these four relations,
$${a^2} + bc = bc + {d^2} \Rightarrow {a^2} = {d^2}$$
and $$\,\,b\left( {a + d} \right) = 0 = c\left( {a + d} \right)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow a = - d$$
We can take $$a = 1,b = 0,c = 0,d = - 1$$
as one possible set of values, then
$$A = \left[ {\matrix{ 1 & 0 \cr 0 & { - 1} \cr } } \right]$$
Clearly $$A \ne I\,\,\,$$ and $$\,\,\,\,A \ne - I\,\,$$ and $$\,\,\,A = - 1$$
$$\therefore$$ $$\,\,\,\,\,$$ Statement $$1$$ is true.
Also if $$A \ne I\,\,\,\,\,tr\left( A \right) = 0$$
$$\therefore$$ $$\,\,\,\,\,$$ Statement $$2$$ is false.
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