Let `a` and `b` be two real numbers such that `a > 1, b > 1.` If `A=[(a,0), (0,b)]` , then `lim_(n to oo) A^(-n)` is
a. unit matrix
b. null matrix
c. `2l`
d. none of these

Options

A. unit matrix
B. null matrix
C. `2I`
D. none of these

Answer

B

Explanation

`A=((a,0),(0,b))`
`implies A^(2)=((a,0),(0,b))((a,0),(0,b))=((a^(2),0),(0,b^(2)))`
`implies A^(3)=((a^(2),0),(0,b^(2))) ((a,0),(0,b))=((a^(3),0),(0,b^(3)))`
`implies A^(n)=((a^(n),0),(0,b^(n)))`
`implies (A^(n))^(-1) =1/(a^(n)b^(n)) ((b^(n),0),(0,a^(n)))=((a^(-n),0),(0,b^(-n)))`
`implies lim_(n rarr oo) (A^(n))^(-1)=((0,0),(0,0))` as `a gt 1` and `b gt 1`