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Question

There are two possible value of A in the solution of the matrix equation $open square brackets table row cell 2 A plus 1 end cell cell negative 5 end cell row cell negative 4 end cell A end table close square brackets to the power of negative 1 end exponent$ $open square brackets table row cell A minus 5 end cell B row cell 2 A minus 2 end cell C end table close square brackets$ = $open square brackets table row 14 D row E F end table close square brackets$ , where A, B, C, D, E, F are real numbers. The absolute value of the difference of these two solutions, is:

$\frac{8}{3}$
$\frac{11}{3}$
$\frac{1}{3}$
$\frac{19}{3}$

The correct answer is: $fraction numerator 19 over denominator 3 end fraction$

$open square brackets table row cell 2 A plus 1 end cell cell negative 5 end cell row cell negative 4 end cell A end table close square brackets to the power of negative 1 end exponent$ = $fraction numerator 1 over denominator 2 A squared plus A minus 20 end fraction open square brackets table attributes columnalign center center columnspacing 1em end attributes row A 5 row 4 cell 2 A plus 1 end cell end table close square brackets$
So $fraction numerator 1 over denominator 2 A squared plus A minus 20 end fraction times open square brackets table attributes columnalign center center columnspacing 1em end attributes row straight A 5 row 4 cell 2 straight A plus 1 end cell end table close square brackets open square brackets table attributes columnalign center center columnspacing 1em end attributes row cell A minus 5 end cell B row cell 2 A minus 2 end cell C end table close square brackets$
= $open square brackets table attributes columnalign center center columnspacing 1em end attributes row 14 D row E F end table close square brackets$ $fraction numerator 1 over denominator 2 A to the power of 2 end exponent plus A minus 20 end fraction$ $open square brackets table row cell A left parenthesis A minus 5 right parenthesis plus 5 left parenthesis 2 A minus 2 right parenthesis end cell cell A B plus 5 C end cell row cell 4 left parenthesis A minus 5 right parenthesis plus left parenthesis 2 A plus 1 right parenthesis left parenthesis 2 A minus 2 right parenthesis end cell cell 4 B plus C left parenthesis 2 A plus 1 right parenthesis end cell end table close square brackets$
= $open square brackets table row 14 D row E F end table close square brackets$
so $fraction numerator A to the power of 2 end exponent plus 5 A minus 10 over denominator 2 A to the power of 2 end exponent plus A minus 20 end fraction$ = 14 $rightwards double arrow$ A = 3, $fraction numerator negative 10 over denominator 3 end fraction$

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