Scalar multiplication

Definition

Scalar multiplication involves multiplying each entry of a matrix by a constant value, known as the scalar. This operation results in a new matrix where each element is the product of the original element and the scalar.

5 Must Know Facts For Your Next Test

1. Scalar multiplication is performed element-wise across the entire matrix.
2. If $c$ is a scalar and $A$ is an $m \times n$ matrix, then $cA$ produces another $m \times n$ matrix.
3. The distributive property applies: $c(A + B) = cA + cB$, where $A$ and $B$ are matrices of the same dimensions.
4. The associative property with respect to scalars holds: $(cd)A = c(dA)$, where $c$ and $d$ are scalars.
5. Multiplying a matrix by the scalar zero results in a zero matrix, where all elements are zero.

Review Questions

• What happens to each element of a matrix when it undergoes scalar multiplication?
• How does scalar multiplication affect the dimensions of an original matrix?
• Explain how the distributive property works in terms of scalar multiplication with matrices.

Related terms

Matrix Addition:

An operation where corresponding elements of two matrices of the same dimensions are added together to form a new matrix.

Zero Matrix: A matrix in which all elements are zeros. It acts as an additive identity in matrix addition.

Matrix Multiplication:

An operation that takes two matrices and produces another matrix by performing dot products between rows and columns.