Question

Find the adjoint and inverse if exists of the following matrices

costita sintita 0
-sintita costita 0
0 0 1​

Answer 1

Solution step by step,
Let A be the given 3x3 matrix,
A= a11 a12 a13
a21 a22 a23
a31 a32 a33

a11= + costita 0 = costita
0 1

a12=- -sintita 0 = + sintita
0 1

a13=+ -sintita Costita = 0
0 0

a21=- sintita. 0 = - sintita
0. 1

a22=+. costita. 0 = costita
0. 1

a23=- costita. Sintita = 0
0. 0

a31=+. sintita. 0 =. 0
costita 0

a32=- costita. 0. =. 0
-sintita. 0

a33=+. costita. sintita. = cos^2tita+sin^2tita
-sintita. costita. = i.e 1


Adjoint of A = costita. sintita. 0
-sintita. costita. 0
0. 0. 1
Now, For inverse.
A inverse = (adjoint of A) / |A|

|A| = costita(costita*1-0*0)-sintita(-sintita*1-0*0)
+0(-sinetita*0-costita*0)
= costita*costita-sintita*(-sintita)
= cos^2tita-sintita^2tita
=. 1

Then, the inverse of A = same as the adjoint
of A (or as question)