MATLAB - Matrix

A matrix is a two-dimensional array of numbers.
In MATLAB, you create a matrix by entering elements in each row as comma or space delimited numbers and using semicolons to mark the end of each row.
For example, let us create a 4-by-5 matrix a −

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8]

MATLAB will execute the above statement and return the following result −

a =
     1     2     3     4     5
     2     3     4     5     6
     3     4     5     6     7
     4     5     6     7     8

Referencing the Elements of a Matrix

To reference an element in the m^th row and n^th column, of a matrix mx, we write −

mx(m, n);

For example, to refer to the element in the 2^nd row and 5^th column, of the matrix a, as created in the last section, we type −

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
a(2,5)

MATLAB will execute the above statement and return the following result −

ans =  6

To reference all the elements in the m^th column we type A(:,m).
Let us create a column vector v, from the elements of the 4^th row of the matrix a:

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
v = a(:,4)

MATLAB will execute the above statement and return the following result −

v =
     4
     5
     6
     7

You can also select the elements in the m^th through n^th columns, for this we write −

a(:,m:n)

Let us create a smaller matrix taking the elements from the second and third columns −

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
a(:, 2:3)

MATLAB will execute the above statement and return the following result −

ans =
     2     3
     3     4
     4     5
     5     6

In the same way, you can create a sub-matrix taking a sub-part of a matrix.

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
a(:, 2:3)

MATLAB will execute the above statement and return the following result −

ans =
     2     3
     3     4
     4     5
     5     6

In the same way, you can create a sub-matrix taking a sub-part of a matrix.
For example, let us create a sub-matrix sa taking the inner subpart of a:

3     4     5     
4     5     6     

To do this, write −

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
sa = a(2:3,2:4)

MATLAB will execute the above statement and return the following result −

sa =
     3     4     5
     4     5     6

Deleting a Row or a Column in a Matrix

You can delete an entire row or column of a matrix by assigning an empty set of square braces [] to that row or column. Basically, [] denotes an empty array.
For example, let us delete the fourth row of a −

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
a( 4 , : ) = []

MATLAB will execute the above statement and return the following result −

a =
     1     2     3     4     5
     2     3     4     5     6
     3     4     5     6     7

Next, let us delete the fifth column of a −

a = [ 1 2 3 4 5; 2 3 4 5 6; 3 4 5 6 7; 4 5 6 7 8];
a(: , 5)=[]

MATLAB will execute the above statement and return the following result −

a =
     1     2     3     4
     2     3     4     5
     3     4     5     6
     4     5     6     7

Example

In this example, let us create a 3-by-3 matrix m, then we will copy the second and third rows of this matrix twice to create a 4-by-3 matrix.
Create a script file with the following code −

a = [ 1 2 3 ; 4 5 6; 7 8 9];
new_mat = a([2,3,2,3],:)

When you run the file, it displays the following result −

new_mat =
     4     5     6
     7     8     9
     4     5     6
     7     8     9

Matrix Operations

In this section, let us discuss the following basic and commonly used matrix operations −

• Addition and Subtraction of Matrices
• Division of Matrices
• Scalar Operations of Matrices
• Transpose of a Matrix
• Concatenating Matrices
• Matrix Multiplication
• Determinant of a Matrix
• Inverse of a Matrix