Question

Let `aa n db` be two real numbers such that `a >1,b > 1.` If `A=(a0 0b)` , then `(lim)_(nvecoo)A^(-n)` is a. unit matrix b. null matrix c. `2l` d. none of these
A. unit matrix
B. null matrix
C. `2I`
D. none of these

Answer 1

Correct Answer - B
`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`