Addition and Scalar Multiplication of Matrices
Introduction to Matrices
A matrix is a rectangular array of numbers. The plural form of matrix is matrices. You have encountered matrices before in the context of augmented matrices and coefficient matrices associate with linear systems.
Consider the matrix
The dimension of a matrix is defined as where is the number of rows and is the number of columns. The above matrix is a matrix because there are three rows and four columns.
A column vector in is an matrix. A row vector in is a matrix.
The individual entries in the matrix are identified according to their position. The -entry of a matrix is the entry in the row and column. For example, in matrix above, is called the -entry because it is in the second row and the third column.
We denote the entry in the row and the column of matrix by , and write in terms of its entries as .
Occasionally it will be convenient to talk about columns and rows of a matrix as vectors. We will use the following notation:
A matrix is called a square matrix if it has the same number of rows and columns. If is an square matrix, the entries of the form are said to lie on the main diagonal. For example, if then the main diagonal consists of entries , and .
There are various operations which are done on matrices of appropriate sizes. Matrices can be added to and subtracted from other matrices, multiplied by a scalar, and multiplied by other matrices. We will never divide a matrix by another matrix, but we will see later how multiplication by a matrix inverse (if an inverse exists) plays a similar role to division.
In doing arithmetic with matrices, we often define the action by what happens in terms of the entries (or components) of the matrices. Before looking at these operations in depth, consider a few general definitions.
The Zero Matrix The zero matrix is the matrix having every entry equal to zero.
The zero matrix is denoted by .
Addition of Matrices
Given two matrices of the same dimensions, we can add them together by adding their corresponding entries.
Addition of Matrices Let and be two matrices. Then the sum of matrices and ,
denoted by , is an matrix given by
Find the sum of and , if possible. \begin{equation*} A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 0 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 2 & 3 \\ -6 & 2 & 1 \end{bmatrix} \end{equation*}
Notice that both and are of size . Since and are of the same size, addition is
possible. \begin{align*} A + B &= \begin{bmatrix} 1 & 2 & 3 \\ 1 & 0 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 2 & 3 \\ -6 & 2 & 1 \end{bmatrix}\\ &= \begin{bmatrix} 1+5 & 2+2 & 3+3 \\ 1+ -6 & 0+2 & 4+1 \end{bmatrix}\\ &= \begin{bmatrix} 6 & 4 & 6 \\ -5 & 2 & 5 \end{bmatrix} \end{align*}
Going forward, whenever we write it will be assumed that the two matrices are of equal size and addition is possible.
We will prove Properties item:mataddcomm and item:mataddinv. The remaining properties are left as exercises.
- Proof of Property item:mataddcomm:
- The -entry of is given by
The -entry of is given by
Since , for all , , we conclude that .
- Proof of Property item:mataddinv:
- Let be defined by Then .
You will recognize the zero matrix of Theorem th:propertiesofadditionitem:mataddid as the zero matrix of Definition def:zeromatrix.
Scalar Multiplication of Matrices
When a matrix is multiplied by a scalar, the new matrix is obtained by multiplying every entry of the original matrix by the given scalar.
Properties of Scalar Multiplication Let be matrices, and be scalars. Then, the
following properties properties of scalar multiplication hold.
- (a)
- Distributive Law over Matrix Addition \begin{equation*} k \left ( A+B\right ) =k A+ kB \end{equation*}
- (b)
- Distributive Law over Scalar Addition \begin{equation*} \left ( k +p \right ) A= k A+p A \end{equation*}
- (c)
- Associative Law for Scalar Multiplication \begin{equation*} k \left ( p A\right ) = \left ( k p \right ) A \end{equation*}
- (d)
- Multiplication by \begin{equation*} 1A=A \end{equation*}
The proof of this theorem is similar to the proof of Theorem th:propertiesofaddition and is left as an exercise.
Practice Problems
Text Source
The text in this section is an adaptation of Section 2.1 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)
Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, p. 53-58.
2024-09-06 01:54:55