Question

Matrix `A` such that `A^2=2A-I ,w h e r eI` is the identity matrix, the for `ngeq2. A^n` is equal to `2^(n-1)A-(n-1)l` b. `2^(n-1)A-I` c. `n A-(n-1)l` d. `n A-I`
A. `2^(n-1) A-(n-1)I`
B. `2^(n-1) A-I`
C. `nA-(n-1)I`
D. `nA-I`

Answer 1

Correct Answer - C
Given, `A^(2)=2A-I`
Now, `A^(3)=A(A^(2))`
`=A (2A-I)`
`=2A^(2)-A`
`=2(2A-I)-A`
`=3A-2I`
`A^(4)=A(A^(3))`
`=A(3A-2I)`
`=3A^(2)-2A`
`=3(2A-I)-2A`
`=4A-3I`
Following this, we can say `A^(n)=nA-(n-1)I`.