If (T = begin{pmatrix} -2 & -5 \ 3 & 8 end{pmatrix}), find (T^{-1}), the inverse of T.
-
A.
(begin{pmatrix} -8 & -5 \ 3 & 2 end{pmatrix}) -
B.
(begin{pmatrix} -8 & -5 \ 3 & -2 end{pmatrix}) -
C.
(begin{pmatrix} -8 & -5 \ -3 & 2 end{pmatrix}) -
D.
(begin{pmatrix} -8 & -5 \ -3 & -2 end{pmatrix})
Correct Answer: Option A
Explanation
Let (begin{pmatrix} a & b \ c & d end{pmatrix} = T^{-1})
(T . T^{-1} = I)
(begin{pmatrix} -2 & -5 \ 3 & 8 end{pmatrix}begin{pmatrix} a & b \ c & d end{pmatrix} = begin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix})
(implies -2a – 5c = 1)
(-2b – 5d = 0 implies b = frac{-5d}{2})
(3a + 8c = 0 implies a = frac{-8c}{3})
(3b + 8d = 1)
(-2(frac{-8c}{3}) – 5c = frac{16c}{3} – 5c = frac{c}{3} = 1 implies c = 3)
(3(frac{-5d}{2}) + 8d = frac{-15d}{2} + 8d = frac{d}{2} = 1 implies d = 2)
(b = frac{-5 times 2}{2} = -5)
(a = frac{-8 times 3}{3} = -8)
(therefore T^{-1} = begin{pmatrix} -8 & -5 \ 3 & 2 end{pmatrix})