Question

How do you multiply $2 \times 3$ and $2 \times 2$ matrices?

Answer 1

Hint: We know that when we wish to multiply two matrices, we must first multiply the row and column since the resultant matrix will have an effect on the row and column count of our resultant matrix. So, when we multiply two matrices, we first multiply their row and column.

Complete step by step solution:
We know that matrices are used to represent coefficients in a system of linear equations. Elements of matrices are the numbers of functions in the array. The rows of the matrix refer to the horizontal lines of elements while the columns of matrices refer to the vertical lines of elements. If any matrix ‘m’ rows and ‘n’ columns, then the order of the matrix is $m \times n$
Now suppose that $A = \begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix}_{2 \times 3}$ and $B = \begin{bmatrix} 7 & 2 \\ 3 & 9 \\ \end{bmatrix}_{2 \times 2}$
Now we can see that we have two matrices in which we have two rows and three columns in the first matrix and in the second matrix we have two rows and two columns. Both matrices are of different sizes. Also, we know that matrices of different sizes can only be multiplied if and only if the number of columns is equal to the number of rows of the second matrix.

Therefore, we cannot multiply a $2 \times 3$ and $2 \times 2$ matrices.

Note: One thing that students always keep in mind is that when multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.