Question

Suppose $A$ is a 3 × 3 matrix consisting of integer entries that are chosen at random from the set $\{–1000,–999,\dots ,999,1000\}$. Let $P$ be the probability that either $A^2=–I$ or $A$ is diagonal matrix, where $I$ is the 3 × 3 identity matrix. Then

• A. $P<\frac{1}{{10}^{18}}$
• B. $P=\frac{1}{{10}^{18}}$
• C. $\frac{5^2}{{10}^{18}}\le P\le \frac{5^3}{{10}^{18}}$
• D. $P\ge \frac{5^4}{{10}^{18}}$

Answer 1

The correct option is A $P<\frac{1}{{10}^{18}}$

Given :
$A^2=-I$
Taking determinant,
$|A{|}^2=-1$ which is not possible so,
$A$ is diagonal matrix
Now the total possible matrix $={2001}^9$
Number of diagonal matrix $={2001}^3$
So, $P=\frac{{2001}^3}{{2001}^9}=\frac{1}{{2001}^6}$
$\therefore P<\frac{1}{{2000}^6}<\frac{1}{{1000}^6}<\frac{1}{{10}^{18}}$