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 requires that the number of columns in the first matrix matches the number of rows in the second matrix.

5 Must Know Facts For Your Next Test

1. Matrix multiplication is not commutative; $AB \neq BA$ generally.
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 resulting product matrix is calculated as a dot product of the corresponding row from the first matrix and column from the second matrix.
4. Multiplying by an identity matrix ($I$) leaves a matrix unchanged: $AI = A$ and $IA = A$.
5. The distributive properties hold for matrices: $A(B + C) = AB + AC$.

Review Questions

• What must be true about two matrices for their product to be defined?
• Is it always true that $AB = BA$ for any two matrices? Explain why or why not.
• How do you determine the size of the resulting product when multiplying two matrices?

Related terms

Matrix: A rectangular array of numbers arranged in rows and columns used to represent linear transformations or systems of equations.

Identity Matrix: A square matrix with ones on the diagonal and zeros elsewhere, denoted as I, which acts as a multiplicative identity for matrices.

$\text{Dot Product}$: $\text{An operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.}$