Wykazać własności macierzy

[kws]

Weźmy macierze \(\displaystyle{ A,B \mathcal{M}_{m,n}}\) oraz \(\displaystyle{ a,b,c R}\) Wykaż, że

a) \(\displaystyle{ c(aA+bB) = (ca)A+(cb)B,}\)
b) \(\displaystyle{ -aA=(-aA)=a(-A),}\)
c) \(\displaystyle{ (aA)^T=aA^T}\).

[soku11]

c)
\(\displaystyle{ A=\left[\begin{array}{cccc}
a_{0\ 0}&a_{0\ 1}&...&a_{0\ m}\\
a_{1\ 0}&a_{1\ 1}&...&a_{1\ m}\\
...&...&...&....\\
a_{n\ 0}&a_{n\ 1}&...&a_{n\ m}\\
\end{array}\right] \\}\)
\(\displaystyle{ aA=
ft[\begin{array}{cccc}
a\cdot a_{0\ 0}&a\cdot a_{0\ 1}&...&a\cdot a_{0\ m}\\
a\cdot a_{1\ 0}&a\cdot a_{1\ 1}&...&a\cdot a_{1\ m}\\
...&...&...&....\\
a\cdot a_{n\ 0}&a\cdot a_{n\ 1}&...&a\cdot a_{n\ m}\\
\end{array}\right] \\}\)

\(\displaystyle{ (aA)^T=
ft[\begin{array}{cccc}
a\cdot a_{0\ 0}&a\cdot a_{1\ 0}&...&a\cdot a_{n\ 0}\\
a\cdot a_{0\ 1}&a\cdot a_{1\ 1}&...&a\cdot a_{1\ m}\\
...&...&...&....\\
a\cdot a_{0\ m}&a\cdot a_{1\ m}&...&a\cdot a_{m\ n}\\
\end{array}\right]
=}\)

\(\displaystyle{ a\left[\begin{array}{cccc}
a_{0\ 0}&a_{1\ 0}&...&a_{n\ 0}\\
a_{0\ 1}&a_{1\ 1}&...&a_{1\ m}\\
...&...&...&....\\
a_{0\ m}&a_{1\ m}&...&a_{m\ n}
\end{array}\right]=
aA^T}\)

POZDRO

[kws]

dzieki, a moglby ktos jeszcze zrobic a i b ?