Answer:
AB = [tex]\left[\begin{array}{ccc}-3&4&6\\-6&3&5\\5&0&-4\end{array}\right][/tex]
Each column of AB is written as a linear combination of columns of Matrix A in the explanation below.
Step-by-step explanation:
A = [tex]\left[\begin{array}{ccc}-2&2&1\\-3&1&1\\2&0&-1\end{array}\right][/tex]
B= [tex]\left[\begin{array}{ccc}2&-1&0\\1&2&1\\-1&-2&4\end{array}\right][/tex]
We need to write each column of AB as a linear combination of the columns of A so we will multiply each column of A with each column element of B to get the column of AB. So,
AB Column 1 = 2 * [tex]\left[\begin{array}{ccc}-2\\-3\\2\end{array}\right][/tex] + 1 [tex]\left[\begin{array}{ccc}2\\1\\0\end{array}\right][/tex] + (-1) [tex]\left[\begin{array}{ccc}1\\1\\-1\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}-3\\-6\\5\end{array}\right][/tex]
AB Column 2 = (-1)[tex]\left[\begin{array}{ccc}-2\\-3\\2\end{array}\right][/tex] + 2[tex]\left[\begin{array}{ccc}2\\1\\0\end{array}\right][/tex] + (-2)[tex]\left[\begin{array}{ccc}1\\1\\-1\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}4\\3\\0\end{array}\right][/tex]
AB Column 3 = (0)[tex]\left[\begin{array}{ccc}-2\\-3\\2\end{array}\right][/tex] + (1)[tex]\left[\begin{array}{ccc}2\\1\\0\end{array}\right][/tex] + 4[tex]\left[\begin{array}{ccc}1\\1\\-1\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}6\\5\\-4\end{array}\right][/tex]
Finally, we can combine all three columns of AB to form the 3x3 matrix AB.
AB = [tex]\left[\begin{array}{ccc}-3&4&6\\-6&3&5\\5&0&-4\end{array}\right][/tex]
