Matrix addition

This lecture introduces matrix addition, one of the basic algebraic operations that can be performed on matrices.

Table of contents

Definition
Properties of matrix addition
Solved exercises

Definition

Two matrices can be added together if and only if they have the same dimension. Their sum is obtained by summing each element of one matrix to the corresponding element of the other matrix.

The following is a formal definition.

Definition Let $A$ and $B$ be two $L \times K$ matrices. Their sum $A + B$ is another $L \times K$ matrix such that its $(l, k)$ -th element is equal to the sum of the $(l, k)$ -th element of $A$ and the $(l, k)$ -th element of $B$ , for all $k$ and $l$ satisfying $1 \leq k \leq K$ and $1 \leq l \leq L$ .

The following example shows how matrix addition is performed.

Example Let $A$ and $B$ be two $3 \times 2$ matrices

A = 3 \frac{1}{3} 0 210 B = 0 \frac{2}{3} 0 101

Their sum is

A + B = 3 + 0 \frac{1}{3} + \frac{2}{3} 0 + 0 2 + 1 1 + 0 0 + 1 = 310311

Remember that column vectors and row vectors are also matrices. As a consequence, they can be summed in the same way, as shown by the following example.

Example Let $A$ and $B$ be two $3 \times 1$ column vectors

A = 200 B = 103

Their sum is

A + B = 2 + 1 0 + 0 0 + 3 = 303

Properties of matrix addition

Matrix addition enjoys properties that are similar to those enjoyed by the more familiar addition of real numbers.

Proposition (commutative property) Matrix addition is commutative, that is,

A + B = B + A

for any matrices $A$ and $B$ such that the above additions are meaningfully defined.

Proof

This is an immediate consequence of the fact that the commutative property applies to sums of scalars, and therefore to the element-by-element sums that are performed when carrying out matrix addition.

Proposition (associative property) Matrix addition is associative, that is,

A + (B + C) = (A + B) + C

for any matrices $A$ , $B$ and $C$ such that the above additions are meaningfully defined.

Proof

This is an immediate consequence of the fact that the associative property applies to sums of scalars, and therefore to the element-by-element sums that are performed when carrying out matrix addition.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let $A$ and $B$ be $3 \times 3$ matrices defined by

A = 210250101 B = 501730108

Find their sum.

Solution

In order to compute the sum of $A$ and $B$ , we need to sum each element of $A$ with the corresponding element of $B$ :

A + B = 210250101 + 501730108 = 2 + 5 1 + 0 0 + 1 2 + 7 5 + 3 0 + 0 1 + 1 0 + 0 1 + 8 = 711980209

Exercise 2

Let $A$ be the following $3 \times 2$ matrix:

A = 210247

Define the $2 \times 3$ matrix $B$ as follows:

B = 100002

Compute

A + B^{⊤}

where $B^{⊤}$ is the transpose of $B$ .

Solution

The transpose $B^{⊤}$ is a matrix such that its columns are equal to the rows of $B$ :

B^{⊤} = 100002

Now, since $A$ and $B^{⊤}$ have the same dimension, we can compute their sum:

A + B^{⊤} = 210247 + 100002 = 2 + 1 1 + 0 0 + 0 2 + 0 4 + 0 7 + 2 = 310249

Exercise 3

Let $A$ be a $2 \times 2$ matrix defined by

A = 2011

Show that the sum of $A$ and its transpose is a symmetric matrix.

Solution

The transpose of $A$ is

A^{⊤} = 2101

The sum of $A$ and $A^{⊤}$ is

A + A^{⊤} = 2011 + 2101 = 2 + 2 0 + 1 1 + 0 1 + 1 = 4112

Finally, $A + A^{⊤}$ is symmetric if it is equal to its transpose. The latter is

(A + A^{⊤})^{⊤} = 4112

Thus, the assertion is true.

How to cite

Please cite as:

Taboga, Marco (2021). "Matrix addition", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/matrix-addition.