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

Options

A. more then 2
B. 2
C. 0
D. 1

Answer

A

Explanation

Let `X=((a,b),(c,d))((a,b),(c,d))`
`implies X^(2)=((a^(2)+bc,ab+bd),(ac+cd,bc+d^(2)))`
`implies a^(2)+bc=1` and `ab+bd=1implies b(a+d)=1`
`ac+cd=2implies c(a+d)=2implies 2b=c`
Also,
`bc+d^(2)=3implies d^(2)-a^(2)=2`
`implies (d-a)(a+d)=2 implies d-a=2b`
`a+d=1//b`
`implies 2d=2b+1//b, 2a=1//b-2b`
`d=b+1//2 b, a=1//(2b)-b`
`c=2b`
`implies (b^(2)+ 1/(4b^(2)) +1)+2b^(2)=3`
`implies 3b^(2)+1/(4b^(2))=2`
`implies 12b^(2)-8b^(2)+1=0`
`implies (6b^(2)-1) (2b^(2)-1)=0`
or `b= pm 1/sqrt(6)` or `b = pm 1/sqrt(2)`
Therefore, matrices are
`((0,1//sqrt(2)),(sqrt(2),sqrt(2))), ((0,-1//sqrt(2)),(-sqrt(2),-sqrt(2))),((2//sqrt(6),-1//sqrt(6)),(2//sqrt(6),4sqrt(6)))`