There are two possible values of A in the solution of the matrix equation
`[(2A+1,-5),(-4,A)]^(-1) [(A-5,B),(2A-2,C)]=[(14,D),(E,F)]`
where A, B, C, D, E and F are real numbers. The absolute value of the difference of these two solutions, is

Options

A. `8/3`
B. `19/3`
C. `1/3`
D. `11/3`

Answer

B

Explanation

`A^(-1)=("adj. A")/("det. A")`
`implies [(2A+1,-5),(-4,A)]^(-1)= ([(A,5),(4,2A+1)])/(2A^(2)+A-20)`
`implies 1/(2A^(2)+A-20) [(A,5),(4,2A+1)][(A-5,B),(2A-2,C)]`
`=[(14,D),(E,F)]` (from given equation)
`implies (A^(2)+5A-10)/(2A^(2)+A-20)=14`
`implies 27 A^(2)+9A-270=0`
`implies 3A^(2)+A-30=0`
`implies 3A^(2)-9A+10A-30=0`
`implies (3A+10) (A-3)=0`
`implies A=3` or `-10/3`
therefore, sum of possible values of A is `3+10/3=19/3`.