Let A be a 2 [ ] 2 real matrix with entries from 0, 1 and |A| [ ] 0.

Question

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 :

Accepted Answer

Let A = $$\left[ {\begin{pmatrix} a & b  \
   c & d  \

\end{pmatrix} } \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

Solutions · 1