If A = [0&-1 1&0] then A16 is equal to :

Question

If A = [0&-1 1&0] then A16 is equal to : (A) [0&-1 1&0] (B) [0& 1 1&0] (C) [ - 1 &0 0 1 ] (D) [ 1 0 0 1 ]

Tardigrade Admin · Accepted Answer

We have A = [0&-1 1&0] Now, A2 = A.A = beginpmatrix0&-1 1&0 endpmatrix beginpmatrix0&-1 1&0 endpmatrix = beginpmatrix-1&0 0&-1 endpmatrix = - I where I = beginpmatrix1&0 0&1 endpmatrix is identity matrix (A2)8 = (-I)8 = I Hence, A16 = I

Q. If $A = [01 - 1 0]$ then $A^{16}$ is equal to :

$[01 - 1 0]$

$[0110]$

$[- 1 0 01]$

$[1001]$

We have $A = [01 - 1 0]$
Now, $A^{2} = A . A = (01 - 1 0) (01 - 1 0)$
$= (- 1 0 0 - 1) = - I$
where $I = (1001)$ is identity matrix
$(A^{2})^{8} = (- I)^{8} = I$
Hence, $A^{16} = I$

Q. If A=[01​−10​] then A16 is equal to :

[01​−10​]

[01​10​]

[−10​01​]

[10​01​]