"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 :"

Question

Accepted Answer

"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"

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 :

Solution

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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 :

Solution

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