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

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 and be two matrices. Their sum is another matrix such that its -th element is equal to the sum of the -th element of and the -th element of , for all and satisfying and .

The following example shows how matrix addition is performed.

Example Let and be two matricesTheir sum is

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 and be two column vectorsTheir sum is

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,for any matrices and and 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,for any matrices , and 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.

Below you can find some exercises with explained solutions.

Let and be matrices defined byFind their sum.

Solution

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

Let be the following matrix:Define the matrix as follows:Computewhere is the transpose of .

Solution

The transpose is a matrix such that its columns are equal to the rows of :Now, since and have the same dimension, we can compute their sum:

Let be a matrix defined byShow that the sum of and its transpose is a symmetric matrix.

Solution

The transpose of isThe sum of and is

Finally, is symmetric if it is equal to its transpose. The latter isThus, the assertion is true.

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