Matematică >> matrice și determinanți >> 2
\( \color{orange} C \color{dimgray} \in \textit{M}_{m,p}(\mathbb{C}) \), \( \color{orange} C \color{dimgray} = \color{red} A \color{dimgray} \cdot \color{blue} B \), prin
\( \color{orange} c\color{dimgray}_{ik} = \color{red} a\color{dimgray}_{i1} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{1k} + \color{red} a\color{dimgray}_{i2} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{2k} + ... + \color{red} a\color{dimgray}_{ij} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{jk} + ... + \color{red} a\color{dimgray}_{in} \color{dimgray} \cdot \color{blue} b\color{dimgray}_{nk} \), \( i \in \{1, 2, ..., m \} \), \( k \in \{1, 2, ..., p \} \).
\( A = \begin{pmatrix} \color{tomato}3 & \color{tomato}-2 & \color{tomato}1 & \color{tomato}-5 \\ \color{orangered}-10 & \color{orangered}4 & \color{orangered}0 & \color{orangered}2 \color{dimgray} \end{pmatrix} \)
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:
\( \color{orange} C \color{dimgray} = A \cdot B = \)
\( = \begin{pmatrix} \color{tomato}3 & \color{tomato}-2 & \color{tomato}1 & \color{tomato}-5 \\ \color{orangered}-10 & \color{orangered}4 & \color{orangered}0 & \color{orangered}2 \color{dimgray} \end{pmatrix} \cdot \begin{pmatrix} \color{turquoise}-4 & \color{steelblue}2 & \color{blue}0 \\ \color{turquoise}1 & \color{steelblue}-6 & \color{blue}3 \\ \color{turquoise}7 & \color{steelblue}3 & \color{blue}-2 \\ \color{turquoise}2 & \color{steelblue}1 & \color{blue}5 \end{pmatrix} = \)
\( \tiny = \begin{pmatrix} \color{tomato}3 \color{dimgray} \cdot \color{turquoise}(-4) \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{turquoise}1 \color{dimgray} + \color{tomato}1 \color{dimgray} \cdot \color{turquoise}7 \color{dimgray} + \color{tomato}(-5) \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} & \color{tomato}3 \color{dimgray} \cdot \color{steelblue}2 \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{steelblue}(-6) \color{dimgray} + \color{tomato}1 \color{dimgray} \cdot \color{steelblue}3 \color{dimgray} + \color{tomato}(-5) \color{dimgray} \cdot \color{steelblue}1 \color{dimgray} & \color{tomato}3 \color{dimgray} \cdot \color{blue}0 \color{dimgray} + \color{tomato}(-2) \color{dimgray} \cdot \color{blue}3 \color{dimgray} + \color{tomato}1 \color{dimgray} \cdot \color{blue}(-2) \color{dimgray} + \color{tomato}(-5) \color{dimgray} \cdot \color{blue}5 \color{dimgray} \\ \color{orangered}(-10) \color{dimgray} \cdot \color{turquoise}(-4) \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{turquoise}1 \color{dimgray} + \color{orangered}0 \color{dimgray} \cdot \color{turquoise}7 \color{dimgray} + \color{orangered}2 \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} & \color{orangered}(-10) \color{dimgray} \cdot \color{steelblue}2 \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{steelblue}(-6) \color{dimgray} + \color{orangered}0 \color{dimgray} \cdot \color{steelblue}3 \color{dimgray} + \color{orangered}2 \color{dimgray} \cdot \color{steelblue}1 \color{dimgray} & \color{orangered}(-10) \color{dimgray} \cdot \color{blue}0 \color{dimgray} + \color{orangered}4 \color{dimgray} \cdot \color{blue}3 \color{dimgray} + \color{orangered}0 \color{dimgray} \cdot \color{blue}(-2) \color{dimgray} + \color{orangered}2 \color{dimgray} \cdot \color{blue}5 \color{dimgray} \end{pmatrix} = \)
\( \scriptsize = \begin{pmatrix} -12 -2 +7 -10 & 6 +12 +3 -5 & 0 -5 -2 -25 \\ 40 +4 +0 +4 & -20 -24 +0 +2 & 0 +12 +0 +10 \end{pmatrix} = \)
\( = \begin{pmatrix} -17 & 16 & -32 \\ 48 & -42 & 22 \end{pmatrix} \).
Produsul matricelor
\( A =
\begin{pmatrix}
\color{tomato}-2 & \color{tomato}-7 \\ \color{orangered}-4 & \color{orangered}-2 \\ \color{red}-3 & \color{red}4\end{pmatrix}
\)
și
\( B =
\begin{pmatrix}
\color{turquoise}2 & \color{steelblue}9 & \color{blue}7 \\ \color{turquoise}8 & \color{steelblue}-1 & \color{blue}-3 \\ \end{pmatrix}
\)
este \( A \cdot B = \)
exercițiu nou
Produsul matricelor
\( A =
\begin{pmatrix}
\color{tomato}-2 & \color{tomato}-7 \\ \color{orangered}-4 & \color{orangered}-2 \\ \color{red}-3 & \color{red}4\end{pmatrix}
\)
și
\( B =
\begin{pmatrix}
\color{turquoise}2 & \color{steelblue}9 & \color{blue}7 \\ \color{turquoise}8 & \color{steelblue}-1 & \color{blue}-3 \\ \end{pmatrix}
\)
este \( A \cdot B = \)\( \begin{pmatrix}-60 & -11 & 7 \\ -24 & -34 & -22 \\ 26 & -31 & -33\end{pmatrix} \).
\( C = A \cdot B = \)
\( =
\begin{pmatrix}
\color{tomato}-2 & \color{tomato}-7 \\ \color{orangered}-4 & \color{orangered}-2 \\ \color{red}-3 & \color{red}4\end{pmatrix}
\cdot
\begin{pmatrix}
\color{turquoise}2 & \color{steelblue}9 & \color{blue}7 \\ \color{turquoise}8 & \color{steelblue}-1 & \color{blue}-3 \\ \end{pmatrix}
=
\)
\( \scriptsize =
\begin{pmatrix}
\color{tomato}(-2) \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} + \color{tomato}(-7) \color{dimgray} \cdot \color{turquoise}8 & \color{tomato}(-2) \color{dimgray} \cdot \color{steelblue}9 \color{dimgray} + \color{tomato}(-7) \color{dimgray} \cdot \color{steelblue}(-1) & \color{tomato}(-2) \color{dimgray} \cdot \color{blue}7 \color{dimgray} + \color{tomato}(-7) \color{dimgray} \cdot \color{blue}(-3) \\ \color{orangered}(-4) \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} + \color{orangered}(-2) \color{dimgray} \cdot \color{turquoise}8 & \color{orangered}(-4) \color{dimgray} \cdot \color{steelblue}9 \color{dimgray} + \color{orangered}(-2) \color{dimgray} \cdot \color{steelblue}(-1) & \color{orangered}(-4) \color{dimgray} \cdot \color{blue}7 \color{dimgray} + \color{orangered}(-2) \color{dimgray} \cdot \color{blue}(-3) \\ \color{red}(-3) \color{dimgray} \cdot \color{turquoise}2 \color{dimgray} + \color{red}4 \color{dimgray} \cdot \color{turquoise}8 & \color{red}(-3) \color{dimgray} \cdot \color{steelblue}9 \color{dimgray} + \color{red}4 \color{dimgray} \cdot \color{steelblue}(-1) & \color{red}(-3) \color{dimgray} \cdot \color{blue}7 \color{dimgray} + \color{red}4 \color{dimgray} \cdot \color{blue}(-3)\end{pmatrix}
=
\)
\( \scriptsize =
\begin{pmatrix}
-4 - 56 & -18 + 7 & -14 + 21 \\ -8 - 16 & -36 + 2 & -28 + 6 \\ -6 + 32 & -27 - 4 & -21 - 12\end{pmatrix}
=
\)
\( = \)
\( \begin{pmatrix}-60 & -11 & 7 \\ -24 & -34 & -22 \\ 26 & -31 & -33\end{pmatrix} \).
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