Question
Mathematics
Find the product of the following two matrices.
$\left[\begin{array}{ccc}2& -5& 2\\ \\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 1& -2\\ \\ 1& -1& 0\\ \\ 0& 1& -1\end{array}\right]$[[2,-5,2],[],[1,0,0]][[0,1,-2],[],[1,-1,0],[],[0,1,-1]]
order of 1st matrix order of 2 nd matrix
The order of the first matrix is $2×3$2xx3 (2 rows and 3 columns) and the order of the second matrix is $3×3$3xx3 (3 rows and 3 columns). Since the number of columns in the first matrix matches the number of rows in the second matrix, we can multiply them using matrix multiplication.
$\left[\begin{array}{ccc}2& -5& 2\\ \\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 1& -2\\ \\ 1& -1& 0\\ \\ 0& 1& -1\end{array}\right]=\left[\begin{array}{ccc}0& 9& -9\\ \\ 0& 1& 0\end{array}\right]$[[2,-5,2],[],[1,0,0]][[0,1,-2],[],[1,-1,0],[],[0,1,-1]]=[[0,9,-9],[],[0,1,0]]
$\begin{array}{rl}& \left[\begin{array}{ccc}2& -5& 2\\ \\ 1& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 1& -2\\ \\ 1& -1& 0\\ \\ 0& 1& -1\end{array}\right]\\ \\ & =\left[\begin{array}{ccc}\left(2×0\right)+\left(-5×1\right)+\left(2×0\right)& \left(2×1\right)+\left(-5×-1\right)+\left(2×1\right)& \left(2×-2\right)+\left(-5×0\right)+\left(2×-1\right)\\ \\ \left(1×0\right)+\left(0×1\right)+\left(0×0\right)& \left(1×1\right)+\left(0×-1\right)+\left(0×1\right)& \left(1×-2\right)+\left(0×0\right)+\left(0×-1\right)\end{array}\right]\\ \\ & =\left[\begin{array}{ccc}0& 9& -9\\ \\ 0& 1& 0\end{array}\right].\end{array}${:[[[2,-5,2],[],[1,0,0]][[0,1,-2],[],[1,-1,0],[],[0,1,-1]]],[],[=[[(2xx0)+(-5xx1)+(2xx0),(2xx1)+(-5xx-1)+(2xx1),(2xx-2)+(-5xx0)+(2xx-1)],[],[(1xx0)+(0xx1)+(0xx0),(1xx1)+(0xx-1)+(0xx1),(1xx-2)+(0xx0)+(0xx-1)]]],[],[=[[0,9,-9],[],[0,1,0]].]:}