Suppose A is a `3xx3` matrix consisting of integer entries that are chosen at random from the set `{-1000,999,….,999,1000}`. Let P be the probability that either `A^(2)= -I` or A is diagonal, where I is the `3xx3` identity matrix. Then

Options

A. `P lt (1)/(10^(18))`
B. `P=(1)/(10^(18))`
C. `(5^(2))/(10^(18))lePle(5^(3))/(10^(18))`
D. `P gt (5^(4))/(10^(18))`

Answer

A

Explanation

`A=[]_(3xx3)`
`A^(2)= -I`
Taking det
`|A|^(2)=-1`(Not possible)
`A to `diagonal matrix

`P=((2001)^(3))/((2001)^(9))=(1)/((2001)^(6))`
`P=(1+2000)^(-6)`
`=(2000^(-6))(1+(1)/((2001)))^(-6) " less than " bot `
`=(1)/(2^(6)xx10^(18))(darr)`
`P lt (1)/(10^(18))`