How many different diagonal matrices of order n can be formed which are involuntary ?

Options

A. A. `2^n`
B. B.`2^n-1`
C. C. `2^(n-1)`
D. D. n

Answer

`2^(n)`

Explanation

Matrix A is diagonal matrix.
`:. A`=dia. `(a_(1), a_(2), ..., a_(n))`
`implies A^(2)=` dia. `((a_(1))^(2), (a_(2))^(2),..,(a_(n))^(2))`
since A involuntary, we have
`:. A^(2)=A`
`implies (a_(1))^(2)=a_(1), (a_(2))^(2)=a_(2), ..., (a_(n))^(2)=a_(2),...,(a_(n))^(2)=a_(n)`.
`implies a_(1), a_(2), ..., a_(n)=0, 1`.
Thus, number of required matrix A is `2xx2xx2xx...n` times `=2^(n)`