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Pentru a calcula produsul a două matrice, \( \color{red} A \color{dimgray} \in \textit{M}_{m,n}(\mathbb{C}) \), \( \color{blue} B \color{dimgray} \in \textit{M}_{n,p}(\mathbb{C}) \),
\( \color{orange} C \color{dimgray} \in \textit{M}_{m,p}(\mathbb{C}) \), \( \color{orange} C \color{dimgray} = \color{red} A \color{dimgray} \cdot \color{blue} B \), prin
\( \color{orange} c\color{dimgray}_{ik} = \color{red} a\color{dimgray}_{i1} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{1k} + \color{red} a\color{dimgray}_{i2} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{2k} + ... + \color{red} a\color{dimgray}_{ij} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{jk} + ... + \color{red} a\color{dimgray}_{in} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{nk} \), \( i \in \{1, 2, ..., m \} \), \( k \in \{1, 2, ..., p \} \).

exemple

Produsul matricelor
\( A = \begin{pmatrix} \color{tomato}3 & \color{tomato}-2 & \color{tomato}1 & \color{tomato}-5 \\ \color{orangered}-10 & \color{orangered}4 & \color{orangered}0 & \color{orangered}2 \color{dimgray} \end{pmatrix} \)
și
\( B = \begin{pmatrix} \color{turquoise}-4 & \color{steelblue}2 & \color{blue}0 \\ \color{turquoise}1 & \color{steelblue}-6 & \color{blue}3 \\ \color{turquoise}7 & \color{steelblue}3 & \color{blue}-2 \\ \color{turquoise}2 & \color{steelblue}1 & \color{blue}5 \end{pmatrix} \)

este:
\( \color{orange} C \color{dimgray} = A \cdot B = \)
\( = \begin{pmatrix} \color{tomato}3 & \color{tomato}-2 & \color{tomato}1 & \color{tomato}-5 \\ \color{orangered}-10 & \color{orangered}4 & \color{orangered}0 & \color{orangered}2 \color{dimgray} \end{pmatrix} \cdot \begin{pmatrix} \color{turquoise}-4 & \color{steelblue}2 & \color{blue}0 \\ \color{turquoise}1 & \color{steelblue}-6 & \color{blue}3 \\ \color{turquoise}7 & \color{steelblue}3 & \color{blue}-2 \\ \color{turquoise}2 & \color{steelblue}1 & \color{blue}5 \end{pmatrix} = \)

\( \tiny = \begin{pmatrix} \color{tomato}3 \color{dimgray} \cdot \color{turquoise}(-4) \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{turquoise}1 \color{dimgray} + \color{tomato}1 \color{dimgray} \cdot \color{turquoise}7 \color{dimgray} + \color{tomato}(-5) \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} & \color{tomato}3 \color{dimgray} \cdot \color{steelblue}2 \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{steelblue}(-6) \color{dimgray} + \color{tomato}1 \color{dimgray} \cdot \color{steelblue}3 \color{dimgray} + \color{tomato}(-5) \color{dimgray} \cdot \color{steelblue}1 \color{dimgray} & \color{tomato}3 \color{dimgray} \cdot \color{blue}0 \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{blue}3 \color{dimgray} + \color{tomato}1 \color{dimgray} \cdot \color{blue}(-2) \color{dimgray} + \color{tomato}(-5) \color{dimgray} \cdot \color{blue}5 \color{dimgray} \\ \color{orangered}(-10) \color{dimgray} \cdot \color{turquoise}(-4) \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{turquoise}1 \color{dimgray} + \color{orangered}0 \color{dimgray} \cdot \color{turquoise}7 \color{dimgray} + \color{orangered}2 \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} & \color{orangered}(-10) \color{dimgray} \cdot \color{steelblue}2 \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{steelblue}(-6) \color{dimgray} + \color{orangered}0 \color{dimgray} \cdot \color{steelblue}3 \color{dimgray} + \color{orangered}2 \color{dimgray} \cdot \color{steelblue}1 \color{dimgray} & \color{orangered}(-10) \color{dimgray} \cdot \color{blue}0 \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{blue}3 \color{dimgray} + \color{orangered}0 \color{dimgray} \cdot \color{blue}(-2) \color{dimgray} + \color{orangered}2 \color{dimgray} \cdot \color{blue}5 \color{dimgray} \end{pmatrix} = \)

\( \scriptsize = \begin{pmatrix} -12 -2 +7 -10 & 6 +12 +3 -5 & 0 -5 -2 -25 \\ 40 +4 +0 +4 & -20 -24 +0 +2 & 0 +12 +0 +10 \end{pmatrix} = \)

\( = \begin{pmatrix} -17 & 16 & -32 \\ 48 & -42 & 22 \end{pmatrix} \).

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Produsul matricelor
\( A = \begin{pmatrix} \color{tomato}-2 & \color{tomato}-2 & \color{tomato}10 \\ \color{orangered}-10 & \color{orangered}4 & \color{orangered}-5\end{pmatrix} \) și \( B = \begin{pmatrix} \color{turquoise}-6 & \color{steelblue}7 \\ \color{turquoise}-9 & \color{steelblue}0 \\ \color{turquoise}-8 & \color{steelblue}-1 \\ \end{pmatrix} \)

este \( A \cdot B = \)\( \begin{pmatrix}-50 & -24 \\ 64 & -65\end{pmatrix} \).

\( C = A \cdot B = \)

\( = \begin{pmatrix} \color{tomato}-2 & \color{tomato}-2 & \color{tomato}10 \\ \color{orangered}-10 & \color{orangered}4 & \color{orangered}-5\end{pmatrix} \cdot \begin{pmatrix} \color{turquoise}-6 & \color{steelblue}7 \\ \color{turquoise}-9 & \color{steelblue}0 \\ \color{turquoise}-8 & \color{steelblue}-1 \\ \end{pmatrix} = \)

\( \scriptsize = \begin{pmatrix} \color{tomato}(-2) \color{dimgray} \cdot \color{turquoise}(-6) \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{turquoise}(-9) \color{dimgray} + \color{tomato}10 \color{dimgray} \cdot \color{turquoise}(-8) & \color{tomato}(-2) \color{dimgray} \cdot \color{steelblue}7 \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{steelblue}0 \color{dimgray} + \color{tomato}10 \color{dimgray} \cdot \color{steelblue}(-1) \\ \color{orangered}(-10) \color{dimgray} \cdot \color{turquoise}(-6) \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{turquoise}(-9) \color{dimgray} + \color{orangered}(-5) \color{dimgray} \cdot \color{turquoise}(-8) & \color{orangered}(-10) \color{dimgray} \cdot \color{steelblue}7 \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{steelblue}0 \color{dimgray} + \color{orangered}(-5) \color{dimgray} \cdot \color{steelblue}(-1)\end{pmatrix} = \)

\( \scriptsize = \begin{pmatrix} 12 + 18 - 80 & -14 + 0 - 10 \\ 60 - 36 + 40 & -70 + 0 + 5\end{pmatrix} = \)

\( = \) \( \begin{pmatrix}-50 & -24 \\ 64 & -65\end{pmatrix} \).

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Întregul fișier .tex
\documentclass[12pt, a4paper, twoside]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{enumerate} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{geometry} \geometry{ a4paper, total={210mm,297mm}, left=20mm, right=15mm, top=20mm, bottom=10mm, } \usepackage{hyperref} \hypersetup{ hidelinks} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \rhead{\href{http://www.pro-matematica.ro}{pro-matematica.ro}} \usepackage[romanian]{babel} \usepackage{combelow} \usepackage{newunicodechar} \newunicodechar{Ș}{\cb{S}} \newunicodechar{ș}{\cb{s}} \newunicodechar{Ț}{\cb{T}} \newunicodechar{ț}{\cb{t}} \newunicodechar{Ă}{\v{A}} \newunicodechar{ă}{\v{a}} \newunicodechar{Â}{\^{A}} \newunicodechar{â}{\^{a}} \newunicodechar{Î}{\^{I}} \newunicodechar{î}{\^{i}} \usepackage{xlop} \usepackage{cancel} \begin{document} Calculați produsul matricelor \( A = \begin{pmatrix} -2 & -2 & 10 \\ -10 & 4 & -5\end{pmatrix} \) și \( B = \begin{pmatrix} -6 & 7 \\ -9 & 0 \\ -8 & -1 \\ \end{pmatrix} \).\( \newline \) Soluție:\( \newline \) \( C = A \cdot B = \) \( = \begin{pmatrix} -2 & -2 & 10 \\ -10 & 4 & -5\end{pmatrix} \cdot \begin{pmatrix} -6 & 7 \\ -9 & 0 \\ -8 & -1 \\ \end{pmatrix} = \) \( \newline \) \( \scriptsize = \begin{pmatrix} (-2) \cdot (-6) + (-2) \cdot (-9) + 10 \cdot (-8) & (-2) \cdot 7 + (-2) \cdot 0 + 10 \cdot (-1) \\ (-10) \cdot (-6) + 4 \cdot (-9) + (-5) \cdot (-8) & (-10) \cdot 7 + 4 \cdot 0 + (-5) \cdot (-1)\end{pmatrix} = \) \( \newline \) \( \scriptsize = \begin{pmatrix} 12 + 18 - 80 & -14 + 0 - 10 \\ 60 - 36 + 40 & -70 + 0 + 5\end{pmatrix} = \) \( \newline \) \( = \) \( \begin{pmatrix}-50 & -24 \\ 64 & -65\end{pmatrix} \). \vspace{2mm} \end{document}

Doar problema în format .tex
Calculați produsul matricelor \( A = \begin{pmatrix} -2 & -2 & 10 \\ -10 & 4 & -5\end{pmatrix} \) și \( B = \begin{pmatrix} -6 & 7 \\ -9 & 0 \\ -8 & -1 \\ \end{pmatrix} \).\( \newline \) Soluție:\( \newline \) \( C = A \cdot B = \) \( = \begin{pmatrix} -2 & -2 & 10 \\ -10 & 4 & -5\end{pmatrix} \cdot \begin{pmatrix} -6 & 7 \\ -9 & 0 \\ -8 & -1 \\ \end{pmatrix} = \) \( \newline \) \( \scriptsize = \begin{pmatrix} (-2) \cdot (-6) + (-2) \cdot (-9) + 10 \cdot (-8) & (-2) \cdot 7 + (-2) \cdot 0 + 10 \cdot (-1) \\ (-10) \cdot (-6) + 4 \cdot (-9) + (-5) \cdot (-8) & (-10) \cdot 7 + 4 \cdot 0 + (-5) \cdot (-1)\end{pmatrix} = \) \( \newline \) \( \scriptsize = \begin{pmatrix} 12 + 18 - 80 & -14 + 0 - 10 \\ 60 - 36 + 40 & -70 + 0 + 5\end{pmatrix} = \) \( \newline \) \( = \) \( \begin{pmatrix}-50 & -24 \\ 64 & -65\end{pmatrix} \).

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Matematică >> matrice și determinanți >> 2