Multiplication of Matrices

Key Concepts

Matrix multiplication is not commutative, meaning that the order of multiplication matters. In other words, for two matrices A and B, A x B is not necessarily equal to B x A.
For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. If not, the matrices cannot be multiplied.
To multiply matrices, use an element-by-element multiplication and addition process. The element in the i-th row and j-th column of the product matrix is obtained by multiplying corresponding elements from the i-th row of the first matrix and j-th column of the second matrix, then adding these products together.

To avoid mistakes, it is useful to draw grid lines, or use lined paper turned sideways, to easily see the rows and columns.
For each cell of the result matrix, find the dot product of the corresponding row from the first matrix and column from the second matrix.
Remember to keep track of the sign of each element. You could be multiplying a positive value by a negative one, and vice versa.

In case of multiplying by the identity matrix, the original matrix remains unchanged. This is because an identity matrix has 1s on its diagonal and 0s everywhere else, and therefore acts similarly to multiplying a number by 1.
It’s possible for two non-zero matrices to multiply to give the zero matrix. This doesn’t mean that either of the original matrices is a zero matrix, but that their particular combination through multiplication results in a matrix of zeros.

If A is an m x n matrix and B is an n x p matrix, then their product AB is an m x p matrix. The element in the i-th row and j-th column of matrix AB will be the sum of the products of corresponding elements from the i-th row of A and j-th column of B. This is often summarised as (AB)ij = Σk=1^n (Aik * Bkj).

One of the most common errors is to simply multiply corresponding elements in the two matrices without adding up the results or to assume that matrix multiplication is commutative. Always remember, matrix multiplication involves both multiplication and addition and it is not commutative.