Matrix Scalar Multiplication¶

Definition¶

The scalar multiple $\alpha \boldsymbol{A}$ of a matrix $\boldsymbol{A}=(a_{ij})$ is the matrix whose entries are $(\alpha a_{ij})$.

Interpretation¶

Multiplying a matrix by a scalar $\alpha$ multiplies every entry by the same number, so $\alpha\boldsymbol{A}$ has the same size as $\boldsymbol{A}$. For $\alpha=2$ every entry doubles, for $\alpha=-1$ every sign flips, and for $\alpha=0$ the result is the zero matrix.

This is the scalar multiplication of $\mathbb{R}^n$ applied to all entries of the array; together with matrix addition it makes the $m\times n$ matrices a vector space. For the map $\boldsymbol{x}\mapsto\boldsymbol{A}\boldsymbol{x}$ it scales every output, $(\alpha\boldsymbol{A})\boldsymbol{x}=\alpha(\boldsymbol{A}\boldsymbol{x})$$\alpha (\mathbf{A}\mathbf{x})$, which matches the scalar multiple of linear maps. Do not confuse it with the row operation that multiplies a single row: $\alpha\boldsymbol{A}$ scales all rows at once.

Key Results¶

• Scalar multiplication distributes over matrix addition.
• $0\boldsymbol{A}$ is the zero matrix and $1\boldsymbol{A}=\boldsymbol{A}$$\mathbf{A}$.

Examples¶

• $2\begin{pmatrix}1&3\\0&-1\end{pmatrix}=\begin{pmatrix}2&6\\0&-2\end{pmatrix}$$\left(\begin{array}{cc}2& 6\\ 0& -2\end{array}\right)$.
• $-\boldsymbol{A}$ is the additive inverse of $\boldsymbol{A}$.

Prerequisites¶

• Matrix
• Field

Related Concepts¶

• Scalar Multiplication
• Matrix Addition
• Vector Space

Sources¶

• Math II: Linear Algebra · Week 3

Aliases: scalar multiplication of matrices, scalar multiple of a matrix Symbols: alpha, real numbers

Last updated: 20 October 2026 03:02