The number of solutions of the matrix equation `X^2=[1 1 2 3]` is a. more than2 b. `2` c. `0` d. `1`

Answer

4

Explanation

`X=[(a,b),(c,d)]rarrX^(2)=[(a^(2)+bc,ab+bd),(ac+cd,bc+d^(2))]`
`a^(2)+bc=1" …(i)"`
`ab+bd=1" "rArr" "b(a+d)=1`
`ac+cd=2" "c(a+d)=2`
`(b)/(c)=(1)/(2)rarr c=2b`
`bc+d^(2)=3" …(ii)"`
`(ii)-(i)`
`d^(2)-a^(2)=2`
`(d-a)(d+a)=2, (d-a)=2b`
`a+d=(1)/(b),d=b+(1)/(2b),a=(1)/(2b)-b`
`a^(2)+bc=1`
`b^(2)+(1)/(4b^(2))-1+2b^(2)=1`
`3b^(2)+(1)/(4b^(2)-2=0`
`b^(2)=x`
`3x+(1)/(4x)-2x=0rArr 12x^(2)-8x+1=0`
`b^(2)=(1)/(6),b^(2)=(1)/(2)" Four such matrices exist"`