If `A=[[a,b],[c,d]]` (where `b c!=0` ) satisfies the equations `x^2+k=0,t h e n`

Options

A. `a+d=0`
B. `k=-|A|`
C. `k=|A|`
D. none of these

Answer

C

Explanation

We have,
`A^(2)=[(a,b),(c,d)][(a,b),(c,d)]=[(a^(2)+bc,ab+db),(ac+cd,bc+d^(2))]`
As A satisfies `x^(2)+k=0`, we have
`A^(2)+kI=O`
or `[(a^(2)+bc+k,(a+d)b),((a+d)c,bc+d^(2)+k)]=[(0,0),(0,0)]`
or `a^(2)+bc+k=0, bc+d^(2)+k=0`
and `(a+d)b=(a+d) c=0`
As `bc ne 0, b ne 0, c ne 0`, so
`a+d=0`
or `a=-d`
Also,
`k=-(a^(2)+bc)`
`=-(d^(2)+bc)`
`=-((-ad)+bc)`
`=|A|`