Question

How do you multiply a $3 \times 2$ matrix and a $2 \times 2$ matrix?

Answer 1

Hint:
We know that when we want to multiply two matrices then we have to check out the column and rows because the resultant matrix will affect our resultant matrix row, column count on it so, when we multiply two matrix then we first multiply row and column.

Complete step by step solution:
Let – we have two matrices $A$ and $B$ then we can say that
If $A = {[{a_{ij}}]_{m \times n}}$ and $B = {[{b_{ij}}]_{n \times p}}$ then we can say that $A \times B = C$ where the value of $C$ is
Now the matrices $C = {[{c_{ij}}]_{m \times p}}$
Where ${c_{ij}} = \sum\limits_{j = 1}^m {{a_{ij}}{b_{jk}}} = {a_{i1}}{b_{1k}} + {a_{i2}}{b_{2k}} + ........ + {a_{im}}{b_{mk}}$
Let's – take an example which help you to understand very clearly
Let – we have two matrices $A$ and $B$ then we can say that
Suppose we have $A = {\left( {\begin{array}{*{20}{c}}
  1&2&{_{}} \\
  3&4&{} \\
  5&1&{}
\end{array}} \right)_{3 \times 2}}$ and $B = {\left( {\begin{array}{*{20}{c}}
  3&4 \\
  1&2
\end{array}} \right)_{2 \times 2}}$
Now we can see that we have two matrices in which we have three rows and two columns in the first matrix and in the second matrix we have two rows and two columns.

So after multiplication of $A$ and $B$ we will get a matrix in which we have three rows and two columns.

Now we will multiply them then we will get $A \times B = C$ so now we will multiply our matrices
$ = {\left( {\begin{array}{*{20}{c}}
  1&2&{_{}} \\
  3&4&{} \\
  5&1&{}
\end{array}} \right)_{3 \times 2}} \times \;\;\;{\left( {\begin{array}{*{20}{c}}
  3&4 \\
  1&2
\end{array}} \right)_{2 \times 2}}$

After multiplication we will get
$ = {\left( {\begin{array}{*{20}{c}}
  {1 \times 3 + 2 \times 1}&{1 \times 4 + 2 \times 2}&{_{}} \\
  {3 \times 3 + 4 \times 1}&{3 \times 4 + 4 \times 2}&{} \\
  {5 \times 3 + 1 \times 1}&{5 \times 4 + 1 \times 2}&{}
\end{array}} \right)_{3 \times 2}}$
Now after completely solving it we will get

$ = {\left( {\begin{array}{*{20}{c}}
  {3 + 2}&{4 + 4}&{_{}} \\
  {9 + 4}&{12 + 8}&{} \\
  {15 + 1}&{20 + 2}&{}
\end{array}} \right)_{3 \times 2}}$
Now after adding all values in it we will get
$ = {\left( {\begin{array}{*{20}{c}}
  5&{16}&{_{}} \\
  {13}&{20}&{} \\
  {16}&{22}&{}
\end{array}} \right)_{3 \times 2}}$
Which is our required matrices

Therefore the multiplication of matrices $A = {\left( {\begin{array}{*{20}{c}}
  1&2&{_{}} \\
  3&4&{} \\
  5&1&{}
\end{array}} \right)_{3 \times 2}}$ and $B = {\left( {\begin{array}{*{20}{c}}
  3&4 \\
  1&2
\end{array}} \right)_{2 \times 2}}$is the $C = {\left( {\begin{array}{*{20}{c}}
  5&{16}&{_{}} \\
  {13}&{20}&{} \\
  {16}&{22}&{}
\end{array}} \right)_{3 \times 2}}$
Now we have understood how to multiply two matrices of $3 \times 2$ and $2 \times 2$.

Note:
Always remember one thing that if you want to multiply two matrices then in both matrices one matrix number of columns must be equal to the second matrix number of rows.