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Special Matrices

Zero Matrix

A zero matrix is a matrix whose entries are all zero, for example

A zero matrix can be of any size but all the entries in the matrix must be zero for it qualify as a zero matrix.

Square Matrix

A square matrix is a matrix with the same number of rows as columns, in other words, its a matrix with size n x n. For example

The above are both square matrices.

Square matrices are important because some of them have determinants and inverses which we'll see in a moment.

Identity Matrix

An identity matrix is a square matrix whose entries are all one along the main diagonal and zero else where. Identity matrices are denoted by I. Sometimes identity matrices are represented by where n is the size of matrix i.e. n x n

The main diagonal of a matrix refers to the diagonal line that can be drawn from the top left corner of the matrix to its bottom right corner. Only square matrices have a main diagonal.

The entries in a matrix A along the main diagonal are represented by the notation

where i represents the row and j represents the column of the matrix A. For example; is A has size 3 x 3, the entries in the following positions would lie on the main diagonal:

where the above represent the entry in the first row and first column, the entry in the second row and second column and the entry in the third row and third column respectively.

Below are a few examples of Identity matrices.

Identity matrices are some of the most useful matrices in algebra because of the unique way in which they interact with other matrices. Here is one of the important properties of identity matrices;

Given a matrix A

and also

In other words, any matrix multiplied by an identity matrix is always equal to the matrix itself. Note however that the rule of matrix multiplication must be followed.

We can prove this by using a matrix A of size 2 x 2 as follows:

Now looking at IA

Another important property of Identity matrices is as follows:

Given a matrix A, if there exist another matrix B such that:

then B is said to be the inverse of A, i.e.

We'll see matrix inverses in a moment.