Dana je matrica A:
1 2
-1 1
te matrica B:
-1 4
7 1
Odredi AB, te BA matrice te inverznu matricu matrice A.

Za početak ćemo izračunati produkt matrica \( AB \) i \( BA \), a zatim ćemo izračunati inverznu matricu matrice \( A \).

### 1. Produkt matrica \( AB \)

Matrica \( A \) je:
\[
A = \begin{pmatrix}
1 & 2 \\
-1 & 1
\end{pmatrix}
\]
Matrica \( B \) je:
\[
B = \begin{pmatrix}
-1 & 4 \\
7 & 1
\end{pmatrix}
\]

Produkt \( AB \) se dobiva množenjem matrice \( A \) s matricom \( B \):
\[
AB = \begin{pmatrix}
1 & 2 \\
-1 & 1
\end{pmatrix}
\begin{pmatrix}
-1 & 4 \\
7 & 1
\end{pmatrix}
\]

Računamo elemente matrice \( AB \):

\[
AB_{11} = 1 \cdot (-1) + 2 \cdot 7 = -1 + 14 = 13
\]
\[
AB_{12} = 1 \cdot 4 + 2 \cdot 1 = 4 + 2 = 6
\]
\[
AB_{21} = -1 \cdot (-1) + 1 \cdot 7 = 1 + 7 = 8
\]
\[
AB_{22} = -1 \cdot 4 + 1 \cdot 1 = -4 + 1 = -3
\]

Dakle, matrica \( AB \) je:
\[
AB = \begin{pmatrix}
13 & 6 \\
8 & -3
\end{pmatrix}
\]

### 2. Produkt matrica \( BA \)

Sad ćemo izračunati produkt \( BA \):

\[
BA = \begin{pmatrix}
-1 & 4 \\
7 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 2 \\
-1 & 1
\end{pmatrix}
\]

Računamo elemente matrice \( BA \):

\[
BA_{11} = -1 \cdot 1 + 4 \cdot (-1) = -1 - 4 = -5
\]
\[
BA_{12} = -1 \cdot 2 + 4 \cdot 1 = -2 + 4 = 2
\]
\[
BA_{21} = 7 \cdot 1 + 1 \cdot (-1) = 7 - 1 = 6
\]
\[
BA_{22} = 7 \cdot 2 + 1 \cdot 1 = 14 + 1 = 15
\]

Dakle, matrica \( BA \) je:
\[
BA = \begin{pmatrix}
-5 & 2 \\
6 & 15
\end{pmatrix}
\]

### 3. Inverzna matrica matrice \( A \)

Inverznu matricu \( A^{-1} \) ćemo izračunati na sljedeći način. Inverzna matrica matrice \( A \) postoji samo ako je determinanta matrice \( A \) različita od nule.

Najprije ćemo izračunati determinantu matrice \( A \):

\[
\text{det}(A) = 1 \cdot 1 - (-1) \cdot 2 = 1 + 2 = 3
\]

Budući da je determinanta različita od nule, matrica \( A \) ima inverznu matricu, koja se računa prema formuli:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\]
Gdje su elementi \( a, b, c, d \) elementi matrice \( A \):
\[
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
U našem slučaju:
\[
A = \begin{pmatrix}
1 & 2 \\
-1 & 1
\end{pmatrix}
\]
Dakle:
\[
A^{-1} = \frac{1}{3} \begin{pmatrix}
1 & -2 \\
1 & 1
\end{pmatrix} =

Explore matrix operations AB and BA and learn how to calculate the inverse matrix of matrix A. This problem covers matrix multiplication and the formula for calculating the inverse matrix. Suitable for advanced high school students and anyone interested in linear algebra.

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