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

Options

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

Answer

A

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)=I`
`implies (a_(1))^(2), (a_(2))^(2),...,(a_(n))^(2)=1`
`implies a_(1), a_(2), ...,a_(n)=1`
Thus, number of required matrix A is `2xx2xx2xx ...n` times `=2^(n)`.