Question-199164

1 Comment

1. 

   KAMIL ADEN KAMIL

   January 30, 2024 at 9:17 am

   Reply

   To find the value of (I – 2A)^-1 + (I + 2A)^-1, we first need to find the inverse of (I – 2A) and (I + 2A).

   Let’s start with finding the inverse of (I – 2A):

   (I – 2A)^-1 = 1/(det(I – 2A)) * adj(I – 2A)

   where det(I – 2A) is the determinant of (I – 2A) and adj(I – 2A) is the adjugate of (I – 2A).

   det(I – 2A) = |(1-2), -2, -4| = -3
   |-1, (3-2), 4|
   |1, -2, (-3-2)|

   adj(I – 2A) = |(3-2), 4| = |1, 4|
   |-2, (-3-2)| |-2, -5|

   So, (I – 2A)^-1 = -1/3 * |1, 4|
   |-2, -5|

   Now, let’s find the inverse of (I + 2A):

   (I + 2A)^-1 = 1/(det(I + 2A)) * adj(I + 2A)

   where det(I + 2A) is the determinant of (I + 2A) and adj(I + 2A) is the adjugate of (I + 2A).

   det(I + 2A) = |(1+4), -2, -4| = 9
   |-1, (3+4), 4|
   |1, -2, (-3+4)|

   adj(I + 2A) = |(3+4), 4| = |7, 4|
   |-2, (-3+4)| |-2, 1|

   So, (I + 2A)^-1 = 1/9 * |7, 4|
   |-2, 1|

   Now we can substitute these values in the original equation:

   (I – 2A)^-1 + (I + 2A)^-1 = (-1/3 * |1, 4|) + (1/9 * |7, 4|)
   |-2, -5| |-2, 1|

   = (-1/3 * 1 + 1/9 * 7) (-1/3 * 4 + 1/9 * 4)
   (-1/3 * -2 + 1/9 * -2) (-1/3 * -5 + 1/9 * 1)

   = (2/9) (-4/9)
   (2/9) (-17/27)

   = |2/9, -4/9|
   |2/9, -17/27|

   Therefore, the value of (I – 2A)^-1 + (I + 2A)^-1 is |2/9, -4/9|
   |2/9, -17/27|.