Question

How do you find the matrix AB if the matrices are given as $ A=\left( \begin{matrix}
   -3 & -7 & -9 \\
   2 & -4 & -1 \\
   4 & 2 & -1 \\
\end{matrix} \right) $ and $ B=\left( \begin{matrix}
   -9 & -1 & 3 \\
   3 & -7 & 3 \\
   4 & 9 & -9 \\
\end{matrix} \right) $ ?

Answer 1

Hint: We start solving the problem by recalling the fact that the multiplication of the two given matrices \[\left( \begin{matrix}
   {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\
   {{a}_{4}} & {{a}_{5}} & {{a}_{6}} \\
   {{a}_{7}} & {{a}_{8}} & {{a}_{9}} \\
\end{matrix} \right)\] and \[\left( \begin{matrix}
   {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\
   {{b}_{4}} & {{b}_{5}} & {{b}_{6}} \\
   {{b}_{7}} & {{b}_{8}} & {{b}_{9}} \\
\end{matrix} \right)\] as\[\left( \begin{matrix}
   \left( {{a}_{1}}\times {{b}_{1}} \right)+\left( {{a}_{2}}\times {{b}_{4}} \right)+\left( {{a}_{3}}\times {{b}_{7}} \right) & \left( {{a}_{1}}\times {{b}_{2}} \right)+\left( {{a}_{2}}\times {{b}_{5}} \right)+\left( {{a}_{3}}\times {{b}_{8}} \right) & \left( {{a}_{1}}\times {{b}_{3}} \right)+\left( {{a}_{2}}\times {{b}_{6}} \right)+\left( {{a}_{3}}\times {{b}_{9}} \right) \\
   \left( {{a}_{4}}\times {{b}_{1}} \right)+\left( {{a}_{5}}\times {{b}_{4}} \right)+\left( {{a}_{6}}\times {{b}_{7}} \right) & \left( {{a}_{4}}\times {{b}_{2}} \right)+\left( {{a}_{5}}\times {{b}_{5}} \right)+\left( {{a}_{6}}\times {{b}_{8}} \right) & \left( {{a}_{4}}\times {{b}_{3}} \right)+\left( {{a}_{5}}\times {{b}_{6}} \right)+\left( {{a}_{6}}\times {{b}_{9}} \right) \\
   \left( {{a}_{7}}\times {{b}_{1}} \right)+\left( {{a}_{8}}\times {{b}_{4}} \right)+\left( {{a}_{9}}\times {{b}_{7}} \right) & \left( {{a}_{7}}\times {{b}_{2}} \right)+\left( {{a}_{8}}\times {{b}_{5}} \right)+\left( {{a}_{9}}\times {{b}_{8}} \right) & \left( {{a}_{7}}\times {{b}_{3}} \right)+\left( {{a}_{8}}\times {{b}_{6}} \right)+\left( {{a}_{9}}\times {{b}_{9}} \right) \\
\end{matrix} \right)\]. We then use this fact for the given matrices A and B to proceed through the problem. We then make the necessary calculations involving multiplication, addition and subtraction operations to get the required multiplication of matrices AB.

Complete step by step answer:
According to the problem, we are asked to find the matrix AB if the matrices are given as $ A=\left( \begin{matrix}
   -3 & -7 & -9 \\
   2 & -4 & -1 \\
   4 & 2 & -1 \\
\end{matrix} \right) $ and $ B=\left( \begin{matrix}
   -9 & -1 & 3 \\
   3 & -7 & 3 \\
   4 & 9 & -9 \\
\end{matrix} \right) $ .
We have given the matrices $ A=\left( \begin{matrix}
   -3 & -7 & -9 \\
   2 & -4 & -1 \\
   4 & 2 & -1 \\
\end{matrix} \right) $ and $ B=\left( \begin{matrix}
   -9 & -1 & 3 \\
   3 & -7 & 3 \\
   4 & 9 & -9 \\
\end{matrix} \right) $ ---(1).
We know that the multiplication of the matrices \[\left( \begin{matrix}
   {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\
   {{a}_{4}} & {{a}_{5}} & {{a}_{6}} \\
   {{a}_{7}} & {{a}_{8}} & {{a}_{9}} \\
\end{matrix} \right)\] and \[\left( \begin{matrix}
   {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\
   {{b}_{4}} & {{b}_{5}} & {{b}_{6}} \\
   {{b}_{7}} & {{b}_{8}} & {{b}_{9}} \\
\end{matrix} \right)\] is \[\left( \begin{matrix}
   \left( {{a}_{1}}\times {{b}_{1}} \right)+\left( {{a}_{2}}\times {{b}_{4}} \right)+\left( {{a}_{3}}\times {{b}_{7}} \right) & \left( {{a}_{1}}\times {{b}_{2}} \right)+\left( {{a}_{2}}\times {{b}_{5}} \right)+\left( {{a}_{3}}\times {{b}_{8}} \right) & \left( {{a}_{1}}\times {{b}_{3}} \right)+\left( {{a}_{2}}\times {{b}_{6}} \right)+\left( {{a}_{3}}\times {{b}_{9}} \right) \\
   \left( {{a}_{4}}\times {{b}_{1}} \right)+\left( {{a}_{5}}\times {{b}_{4}} \right)+\left( {{a}_{6}}\times {{b}_{7}} \right) & \left( {{a}_{4}}\times {{b}_{2}} \right)+\left( {{a}_{5}}\times {{b}_{5}} \right)+\left( {{a}_{6}}\times {{b}_{8}} \right) & \left( {{a}_{4}}\times {{b}_{3}} \right)+\left( {{a}_{5}}\times {{b}_{6}} \right)+\left( {{a}_{6}}\times {{b}_{9}} \right) \\
   \left( {{a}_{7}}\times {{b}_{1}} \right)+\left( {{a}_{8}}\times {{b}_{4}} \right)+\left( {{a}_{9}}\times {{b}_{7}} \right) & \left( {{a}_{7}}\times {{b}_{2}} \right)+\left( {{a}_{8}}\times {{b}_{5}} \right)+\left( {{a}_{9}}\times {{b}_{8}} \right) & \left( {{a}_{7}}\times {{b}_{3}} \right)+\left( {{a}_{8}}\times {{b}_{6}} \right)+\left( {{a}_{9}}\times {{b}_{9}} \right) \\
\end{matrix} \right)\]. Let us use this result to find the multiplication of the given matrices A and B.
So, we get the multiplication of the matrices A and B as\[AB=\left( \begin{matrix}
   \left( -3\times -9 \right)+\left( -7\times 3 \right)+\left( -9\times 4 \right) & \left( -3\times -1 \right)+\left( -7\times -7 \right)+\left( -9\times 9 \right) & \left( -3\times 3 \right)+\left( -7\times 3 \right)+\left( -9\times -9 \right) \\
   \left( 2\times -9 \right)+\left( -4\times 3 \right)+\left( -1\times 4 \right) & \left( 2\times -1 \right)+\left( -4\times -7 \right)+\left( -1\times 9 \right) & \left( 2\times 3 \right)+\left( -4\times 3 \right)+\left( -1\times -9 \right) \\
   \left( 4\times -9 \right)+\left( 2\times 3 \right)+\left( -1\times 4 \right) & \left( 4\times -1 \right)+\left( 2\times -7 \right)+\left( -1\times 9 \right) & \left( 4\times 3 \right)+\left( 2\times 3 \right)+\left( -1\times -9 \right) \\
\end{matrix} \right)\].
Now, let us perform the multiplication, addition, and subtraction operations inside the matrices.
 $ \Rightarrow AB=\left( \begin{matrix}
   \left( 27 \right)+\left( -21 \right)+\left( -36 \right) & \left( 3 \right)+\left( 49 \right)+\left( -81 \right) & \left( -9 \right)+\left( -21 \right)+\left( 81 \right) \\
   \left( -18 \right)+\left( -12 \right)+\left( -4 \right) & \left( -2 \right)+\left( 28 \right)+\left( -9 \right) & \left( 6 \right)+\left( -12 \right)+\left( 9 \right) \\
   \left( -36 \right)+\left( 6 \right)+\left( -4 \right) & \left( -4 \right)+\left( -14 \right)+\left( -9 \right) & \left( 12 \right)+\left( 6 \right)+\left( 9 \right) \\
\end{matrix} \right) $ .
 $ \Rightarrow AB=\left( \begin{matrix}
   27-21-36 & 3+49-81 & -9-21+81 \\
   -18-12-4 & -2+28-9 & 6-12+9 \\
   -36+6-4 & -4-14-9 & 12+6+9 \\
\end{matrix} \right) $ .
 $ \Rightarrow AB=\left( \begin{matrix}
   -30 & -29 & 51 \\
   -34 & 17 & 3 \\
   -34 & -27 & 27 \\
\end{matrix} \right) $ .
So, we have found the multiplication of the matrices $ A=\left( \begin{matrix}
   -3 & -7 & -9 \\
   2 & -4 & -1 \\
   4 & 2 & -1 \\
\end{matrix} \right) $ and $ B=\left( \begin{matrix}
   -9 & -1 & 3 \\
   3 & -7 & 3 \\
   4 & 9 & -9 \\
\end{matrix} \right) $ as $ AB=\left( \begin{matrix}
   -30 & -29 & 51 \\
   -34 & 17 & 3 \\
   -34 & -27 & 27 \\
\end{matrix} \right) $ .
 $ \therefore $ The result of the multiplication of the matrices $ A=\left( \begin{matrix}
   -3 & -7 & -9 \\
   2 & -4 & -1 \\
   4 & 2 & -1 \\
\end{matrix} \right) $ and $ B=\left( \begin{matrix}
   -9 & -1 & 3 \\
   3 & -7 & 3 \\
   4 & 9 & -9 \\
\end{matrix} \right) $ is $ AB=\left( \begin{matrix}
   -30 & -29 & 51 \\
   -34 & 17 & 3 \\
   -34 & -27 & 27 \\
\end{matrix} \right) $ .

Note:
 We should perform each step carefully in order to avoid confusion and calculation mistakes while solving this problem. Whenever we get this type of problem, we first recall the required definition to get the required result for the problem. We can also find the multiplication of the matrices B and A i.e., BA, and then check the relation between the matrices AB and BA. Similarly, we can expect problems to find the trace of the matrices AB and BA.