Video Transcript
Consider the matrices 𝐴 and
𝐵. Find 𝐴𝐵, if possible.
Let’s first establish whether the
multiplication of these two matrices is possible. Matrix 𝐴 has three rows and two
columns, and matrix 𝐵 has two rows and three columns. Seen is the number of columns in
matrix 𝐴 is the same as the number of rows in matrix 𝐵, we know that the resultant
matrix exists. Additionally, we can tell the
dimensions of the resultant matrix by seeing that matrix 𝐴 has three rows and
matrix 𝐵 has three columns. So the resultant matrix will be a
three-by-three matrix. We find the first element of 𝐴𝐵
by multiplying the top row of matrix 𝐴 by the left-hand column of matrix 𝐵. Remember that this is just the same
as finding the dot product of this first row of matrix 𝐴 with the left-hand column
of matrix 𝐵.
That is 11 multiplied by negative
eight add negative two multiplied by negative four. To get the top middle element, we
multiply the top row of matrix 𝐴 with the middle column of matrix 𝐵. That is 11 multiplied by negative
nine add negative two multiplied by eight. To get the top right element, we
multiply the top row of matrix 𝐴 with the right-hand column of matrix 𝐵. That is 11 multiplied by six add
negative two multiplied by nine. We can then find the middle left
component by multiplying the middle row of matrix 𝐴 with the left-hand column of
matrix 𝐵.
We find the middle component by
multiplying the middle row of matrix 𝐴 with the middle column of matrix 𝐵. And we find the middle right
component by multiplying together the middle row of matrix 𝐴 with the right-hand
column of matrix 𝐵. And we follow the same pattern for
the bottom left component, the bottom middle component, and the bottom right
component. We can then simplify each
component. And that gives us our final answer,
which is just as we worked out a three-by-three matrix.
