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Matrix multiplication

If you multiply one matrix that has rows and columns () with another matrix that has rows and columns (), these rules apply,

Rule

Example

When multiplying matrices, the order matters

Although for two variables, for two matrices unless they are both square matrices.

To multiply two matrices, must equal .

For example, you cannot multiply a matrix with a matrix.

The product will have rows and columns.

For example, multiplying a matrix with a will result in a matrix. This also means that multiplying a matrix with one row by a matrix with one column, will result in just a single number.

is the product of row of matrix and column of matrix

For example,

In other words, starting from the first matrix in the product, multiply the first row by each column in the second matrix. Next, multiply the second row by each column in the second matrix. Then, multiply the third row by each column in the second matrix, and so on and so forth.

Each value in the product is calculated as its own product

For example,

A sample problem below,

To double-check your answer, make sure the number final product has the same number of rows as the first matrix, and the same number of columns as the second matrix.