Example:
(a) Find the inverse of the matrix ,
(b) Find the inverse of the matrix
and show that
aA–1 = (a2 +bc + 1 ) I – aA.
(c) Find the matrix
P A Q = R
A = P–1 R Q–1
(d) Let F (x) =
Show that
(i) [F(x)]–1 = F (–x) ; (ii) [G(y)–1] = G(–y) ; (iii) [F(x) G(y)–1] = G (–y) F (–x)
[Solution:
(i) We have, F (x) F (–x)
Also , F (–x) F(x) = I3
F(x) F(–x) = I3 = F (–x) F(x)
Þ F(–x) is the inverse of matrix F(x) i.e. [F(x)]–1 = F (–x)
(ii) we have, G (y) G(–y) =
Similarly, we have
G(–y) G (y) = I3
G(y) G (–y) = I3 = (G –y) G (y)
Þ G(–x) is the inverse of G(x) i.e. [G(x)]–1 = G (–x)
(iii) We have, [F(x) G(y)–1 = [G(y)–1 [F(x)]–1 [ (AB)–1 = B–1 A–1]
= G(–y) F( – x) [Using (i) and (ii)]
(e) If A is an orthogonal non-singular matrix, then prove that A–1 is also an orthogonal matrix.
[Solution : Since A is an orthogonal matrix. Therefore,
AAT = I = ATA
Þ (AAT)–1 = I = (AT A)–1
Þ (AT)–1 A–1 = I = A–1 (AT)–1
Þ (A–1) T A–1 = I = A–1 (A–1)T [ (AT)–1 = (A–1)T]
A–1 is an orthogonal matrix.
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