Product of two matrices

Definition

The product of two matrices is a new matrix obtained by multiplying corresponding entries and summing the results. This operation is only possible when the number of columns in the first matrix equals the number of rows in the second matrix.

5 Must Know Facts For Your Next Test

1. Matrix multiplication is not commutative; $AB \neq BA$ in general.
2. The resulting matrix from multiplying an $m \times n$ matrix by an $n \times p$ matrix will be an $m \times p$ matrix.
3. Each entry in the product matrix is computed as the dot product of a row from the first matrix and a column from the second matrix.
4. If either of the matrices is a zero matrix, the product will also be a zero matrix.
5. Matrix multiplication can be used to solve systems of linear equations by expressing them in terms of matrices.

Review Questions

• What are the conditions necessary for two matrices to be multiplied?
• How do you determine the size of the resulting matrix when multiplying two matrices?
• Explain why matrix multiplication is not commutative with an example.

Related terms

Matrix:

A rectangular array of numbers arranged in rows and columns.

Identity Matrix:

A square matrix with ones on the diagonal and zeros elsewhere; it acts as a multiplicative identity in matrix multiplication.

Transpose:

An operation that flips a matrix over its diagonal, switching its rows with its columns.