Matrix multiplication

Definition

Matrix multiplication is an operation that takes two matrices and produces another matrix. It involves multiplying rows of the first matrix by columns of the second matrix and summing the products.

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 element in the resulting matrix is computed as the dot product of a row from the first matrix with a column from the second matrix.
4. Matrix multiplication is associative: $(AB)C = A(BC)$.
5. To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.

Review Questions

• Why is matrix multiplication not commutative?
• What are the dimensions of the resulting matrix when you multiply a $2 \times 3$ matrix by a $3 \times 4$ matrix?
• Explain how to compute a single element in the result of a matrix multiplication.

Related terms

Identity Matrix:

A square matrix with ones on its diagonal and zeros elsewhere. It acts as a multiplicative identity in matrix operations.

Inverse Matrix:

A square matrix that, when multiplied by its original, yields the identity matrix. Not all matrices have inverses.

Determinant:

A scalar value that can be computed from a square matrix and encodes certain properties of the linear transformation described by the matrix.