If A = [i&0 0&-i], B = [0&-1 1&0] and C= [0&i i&0] then A2 = B2 =C2 is equal to :

Question

If A = [i&0 0&-i], B = [0&-1 1&0] and C= [0&i i&0] then A2 = B2 =C2 is equal to : (A) I2 (B) I (C) - I (D) 2I

Tardigrade Admin · Accepted Answer

Let A = [i&0 0&-i], B = [0&-1 1&0] and C= [0&i i&0] A. A= beginpmatrixi&0 0&-i endpmatrix beginpmatrixi&0 0&-1 endpmatrix = beginpmatrixi2&0 0&i2 endpmatrix = beginpmatrix-1&0 0&-1 endpmatrix = - I and B.B = beginpmatrix0&-1 1&0 endpmatrix beginpmatrix0&-1 1&0 endpmatrix = beginpmatrix-1&0 0&-1 endpmatrix = -I C.C = beginpmatrix0&i i&0 endpmatrix beginpmatrix0&i i&0 endpmatrix = beginpmatrixi2&0 0&i2 endpmatrix = -I Hence, A2 = B2 = C2 = - I

Q. If $A = [i 0 0 - i], B = [01 - 1 0]$ and $C = [0 i i 0]$ then $A^{2} = B^{2} = C^{2}$ is equal to :

$I^{2}$

I

- I

2I

Let $A = [i 0 0 - i], B = [01 - 1 0]$ and $C = [0 i i 0]$
$A . A = (i 0 0 - i) (i 0 0 - 1)$
$= (i^{2} 0 0 i^{2}) = (- 1 0 0 - 1) = - I$
and $B . B = (01 - 1 0) (01 - 1 0)$
$= (- 1 0 0 - 1) = - I$
$C . C = (0 i i 0) (0 i i 0) = (i^{2} 0 0 i^{2}) = - I$
Hence, $A^{2} = B^{2} = C^{2} = - I$

Q. If A=[i0​0−i​],B=[01​−10​] and C=[0i​i0​] then A2=B2=C2 is equal to :

I2

I

- I

2I

Let A=[i0​0−i​],B=[01​−10​] and C=[0i​i0​] A.A=(i0​0−i​)(i0​0−1​) =(i20​0i2​)=(−10​0−1​)=−I and B.B=(01​−10​)(01​−10​) =(−10​0−1​)=−I C.C=(0i​i0​)(0i​i0​)=(i20​0i2​)=−I Hence, A2=B2=C2=−I

Q. If $A = [i 0 0 - i], B = [01 - 1 0]$ and $C = [0 i i 0]$ then $A^{2} = B^{2} = C^{2}$ is equal to :

$I^{2}$

Let $A = [i 0 0 - i], B = [01 - 1 0]$ and $C = [0 i i 0]$
$A . A = (i 0 0 - i) (i 0 0 - 1)$
$= (i^{2} 0 0 i^{2}) = (- 1 0 0 - 1) = - I$
and $B . B = (01 - 1 0) (01 - 1 0)$
$= (- 1 0 0 - 1) = - I$
$C . C = (0 i i 0) (0 i i 0) = (i^{2} 0 0 i^{2}) = - I$
Hence, $A^{2} = B^{2} = C^{2} = - I$