We want to calculate matrix product (DE)F if it is defined. Recall that we can multiply two matrices only if the number of columns of the first matrix is equal to the number of rows of the second matrix. Otherwise, the operation is undefined.
D E F
(
1 & 2 & - 1 0 & 3 & 1 2 & -1 & -2
*
2 & -5 & 0 1 & 0 & -2 3 & 1 & 1
)
*
-3 & 2 -5 & 1 2 & 4
3* 3 3* 3 3* 2
DE
We first need to calculate DE. The dimensions of matrix D are 3* 3, and the dimensions of E are also 3* 3. This means that the number of columns of D is equal to the number of rows of E. Therefore, the multiplication is defined and the dimensions of the product will be 3 * 3.
[
ccc
1 & 2 & - 1 0 & 3 & 1 2 & -1 & -2
]
and
[
cc
2 & -5 & 0 1 & 0 & -2 3 & 1 & 1
]
cc
3* 3 & 3* 3
To multiply matrices, we use a process called the dot product. It follows a pattern of multiplying and adding terms across the rows of the first matrix and down the columns of the second one.
a& b& c d&e&f g&h&i
*
j&k&l m&n&o p&q&r
=
a j + b m + c p&ak + bn + cq&al + bo + cr dj + em + fp&dk + en + fq&dl + eo + fr gj + hm + ip&gk + hn + iq&gl + ho + ir
Let's calculate it!
Finally, to calculate (DE)F, we will multiply the obtained matrix by F. This time the dimensions of the matrices are 3* 3 and 3* 2. This means that the number of columns of DE is equal to the number of rows of F. Therefore, the multiplication is defined.
