标量乘法：闭包性质

The closure property for scalar multiplication is very simple. It just states that if you have a scalar (a real number) $c$ and a matrix $A$ that is size $m\times n$, then when you multiply $A$ by $c$ the resulting matrix is also of size $m\times n$. So $cA$ is also $m\times n$.

Example

Let $c=2$.

And $$A=\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}$$

Then $$cA=2*\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}=\begin{bmatrix}2&4\\6&8\\10&12\end{bmatrix}$$ Notice this matrix $cA$ is also the same size ($2\times3$) as the original matrix $A$.

The Closure Property of Matrix Scalar Multiplication

In this lesson, we will learn about the Closure Property of Matrix Scalar Multiplication.

For all scalar quantities that are rational numbers, you can multiply the scalar quantity by all members of the set (i.e., the values inside the matrix). The Closure Property says that if you multiply a matrix by a scalar quantity, the size of the matrix stays the same.

(Video) Closure Property

by Anusha Rahman

In this video by Anusha Rahman, you will learn about the closure property of scalar multiplication of matrices.

Summary

The Closure Property says that:

• One of the properties of scalar multiplication.
• The size of the matrix product stays the same once a scalar quantity is multiplied by the matrix.

In mathematical terms, this looks like: $$\text{Let A be a m x n matrix.}$$ $$\text{pA = m x n}$$