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^(2)=[[a,b],[c,d]]`
`therefore X^(2)=[[a^(2)+bc,b(a+d)],[c(a+d),bc+d^(2)]]=[[1,1],[2,3]]`
`rArr a^(2) + bc = 1, b(a+d) = 1, `
` c(a+d) = 2 and bc + d^(2) = 3 `
`rArr d^(2) - a^(2) = 2`
`rArr d-a = 2/(d+a) = 2b and d+a = 1/b`
`therefore 2d = 2b + 1/b and 2a = 1/b - 2b`
Also, c= 2b
Now, from `bc+ d^(2) = 3`
`rArr 2b^(2) + ( b+ 1/(2b))^(2) = 3 rArr 3b^(2) + 1/ 4b^(2) - 2 = 0 `
`rArr 12 b^(4) - 8 b^(2) + 1 = 0 `
or `(6b^(2)-1) (2b^(2) -1) = 0`
`rArr b=pm 1/sqrt(6) or b = pm 1/sqrt2`
Therefoer, 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),4/sqrt(6)]]and [[-2/sqrt(6),-1/sqrt(6)],[-2/sqrt(6),-4/sqrt(6)]]`