If `(I-A)^(-1)=I+A+A^2+...+A^7,` then `A^n=O ,` where `n` is

Column I, Column II (I-A)^n is if A is idempotent, p. 2^(n-1)(l-A) (I-A)^n is if A is involuntary, q. I-n A (I-A)^n is if A is nilpotent of index 2, r. A If A is orthogonal, then (A^T)^(-1) , s. I-A

The mean and variance of n observations x_(1),x_(2),x_(3),...x_(n) are 5 and 0 respectively. If sum_(i=1)^(n)x_(i)^(2)=400 , then the value of n is equal to

Let A be an nth-order square matrix and B be its adjoint, then |A B+K I_n| is (where K is a scalar quantity) (|A|+K)^(n-2) b. (|A|+)K^n c. (|A|+K)^(n-1) d. none of these

Prove that (a) (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer. (b) (1+isqrt(3))^n+(1-isqrt(3)^n=2^(n+1)cos((npi)/3) , where n is a positive integer

The value of sum_(n=1)^(13) (i^n+i^(n+1)) , where i =sqrt(-1) equals (A) i (B) i-1 (C) -i (D) 0

Suppose n is a natural number such that |i + 2i^2 + 3i^3 +...... + ni^n|=18sqrt2 where i is the square root of -1 . Then n is

i^(n) + i^(n+1) + i^(n + 2) + i^(n + 3)

Let A=[(0,1),(0,0)] , show that (aI+bA)^(n)=a^(n)I+na^(n-1)bA , where I is the identity matrix of order 2 and n in N .

Show that : (i) {i^(19) + (1/i)^(25)}^(2) = -4 " "(ii) {i^(17) - (1/i)^(34)}^(2) = 2i (iii) {i^(18) + (1/i)^(24)}^(3) = 0 " " (iv) i^n + i^(n+1) + i^(n +2) + i^(n + 3) = 0 , for all n in N .

A data consists of n observations x_(1), x_(2), ..., x_(n). If Sigma_(i=1)^(n) (x_(i) + 1)^(2) = 9n and Sigma_(i=1)^(n) (x_(i) - 1)^(2) = 5n , then the standard deviation of this data is