JEE MAIN - Mathematics (2020 - 2nd September Morning Slot - No. 6)
Let A be a 2 $$ \times $$ 2 real matrix with entries from
{0, 1} and |A|
$$ \ne $$ 0. Consider the following two
statements :
(P) If A $$ \ne $$ I2 , then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,
where I2 denotes 2 $$ \times $$ 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :
(P) If A $$ \ne $$ I2 , then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,
where I2 denotes 2 $$ \times $$ 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :
(P) is true and (Q) is false
Both (P) and (Q) are false
Both (P) and (Q) are true
(P) is false and (Q) is true
Explanation
Let A = $$\left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$, where a, b, c, d $$ \in $$ {0, 1}
$$ \Rightarrow $$ |A| = ad – bc
$$ \therefore $$ ad = 0 or 1 and bc = 0 or 1
So possible values of |A| are 1, 0 or –1
(P) If A $$ \ne $$ I2 then |A| is either 0 or –1
(Q) If |A| = 1 then ad = 1 and bc = 0
$$ \Rightarrow $$ a = d = 1 $$ \Rightarrow $$ Tr(A) = 2
$$ \Rightarrow $$ |A| = ad – bc
$$ \therefore $$ ad = 0 or 1 and bc = 0 or 1
So possible values of |A| are 1, 0 or –1
(P) If A $$ \ne $$ I2 then |A| is either 0 or –1
(Q) If |A| = 1 then ad = 1 and bc = 0
$$ \Rightarrow $$ a = d = 1 $$ \Rightarrow $$ Tr(A) = 2
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