Tolong bantu yang nomer 3,4,5 makasih soal invers matriks persegi

Penyelesaian:

Misal: |A| = determinan 
1/|A| = invers
adj = adjoin
matriks X dan N= [tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right] [/tex]

adj (adjoin) matrik:
[tex]A = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] [/tex]
maka adj dari matriks A = [tex]\left[\begin{array}{ccc}d&-b\\-c&a\end{array}\right] [/tex]

rumus determinan:
[tex]A = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] [/tex]
|A| = ad - bc

rumus invers:
[tex]A = \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] [/tex]
1/|A| = |A| . adj

3. [tex]P = \left[\begin{array}{ccc}2&5\\1&3\end{array}\right] [/tex] , [tex]Q = \left[\begin{array}{ccc}5&4\\1&1\end{array}\right] [/tex]

|A| = 6 - 5 = 1
invers P:
1/|A| = 1 . [tex]\left[\begin{array}{ccc}3&-5\\1&3\end{array}\right] [/tex]
[tex]P^-1 = \left[\begin{array}{ccc}3&-5\\1&2\end{array}\right] [/tex]

[tex]Q = \left[\begin{array}{ccc}5&4\\1&1\end{array}\right] [/tex]

|A| = 5 - 4 = 1
invers Q:
1/|A| = 1 . [tex]\left[\begin{array}{ccc}1&-4\\-1&5\end{array}\right] [/tex]
[tex]Q^-1 = \left[\begin{array}{ccc}1&-4\\-1&5\end{array}\right] [/tex]

P^-1 . Q^-1 = [tex]\left[\begin{array}{ccc}3&-5\\1&2\end{array}\right] . \left[\begin{array}{ccc}1&-4\\-1&5\end{array}\right] = \left[\begin{array}{ccc}8&-27\\-1&4\end{array}\right] [/tex]
|A| = 59 (nilai deteriminan)

4. [tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right] . \left[\begin{array}{ccc}3&2\\2&1\end{array}\right] = \left[\begin{array}{ccc}8&8\\4&5\end{array}\right] [/tex]
lakukan perkalian matriks didapat persamaan berikut lalu eliminasi:
3a + 2b = 8   |x1
2a + b = 8     |x2
---------------  -
a = 8 ; b = -8

3c + 2d = 4   |x1
2c + d  = 5    |x2
---------------  -
c = 6 ; d = -7

maka matriks X = [tex]\left[\begin{array}{ccc}8&-8\\6&-7\end{array}\right] [/tex]

5. [tex]\left[\begin{array}{ccc}3&2\\5&4\end{array}\right] . \left[\begin{array}{ccc}a&b\\c&d\end{array}\right] = \left[\begin{array}{ccc}3&2\\-4&-6\end{array}\right] [/tex]
lalukan perkalian matriks didapat persamaan berikut lalu eliminasi:
3a + 2c = 3   |x2
5a + 4c = -4  |x1
------------------  -
a = 10 ; c = -27/2

3b + 2d = 2
5b + 4d = -6
-----------------  -
b = 10 ; d = -14

maka matriks N = [tex]\left[\begin{array}{ccc}10&10\\ -\frac{27}{2} &-14\end{array}\right] [/tex]