Math Problem Statement
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.
Solution
Za početak ćemo izračunati produkt matrica i , a zatim ćemo izračunati inverznu matricu matrice .
1. Produkt matrica
Matrica je:
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} =Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Inverse Matrix
Formulas
Matrix multiplication
Inverse matrix formula
Theorems
-
Suitable Grade Level
Advanced High School
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