Definition of matrix

A matrix (matrices for plural) is a rectangular array of elements. These elements, called entries of the matrix, are usually numbers. The size or order of a matrix is defined by the number of rows and the number of columns.

For example,

[ 1 2 3 4 5 6 ]  is a matrix of size two-by-three, written as 2×3

[ 1 0 1 3 2 4 ] is a matrix of size three-by-two, written as 3×2

Types of matrices

Square matrix

A square matrix has the same number of rows and columns. For example, 

[ 1 6 1 3 ]  is a square matrix of size 2×2 and [ 1 3 2 0 -8 3 1 5 -1 ]  is a 3×3 square matrix.

Symmetric matrix

A symmetric matrix is a square matrix where when the rows and columns are interchanged, you get the same matrix. For example, if  you interchange the rows and columns of  M= [ 1 3 2 3 -8 3 2 3 -1 ] , you still get matrix M.

Upper triangular matrix

An upper triangular matrix is a square matrix where the entries below the main diagonal are 0. For example, M= [ 1 3 2 0 -2 3 0 0 -1 ]  is an upper triangular matrix.

Lower triangular matrix

A lower triangular matrix is a square matrix where the entries above the main diagonal are 0. For example, M= [ 1 0 0 2 -2 0 3 0 -1 ]  is a lower triangular matrix.

Diagonal matrix

A diagonal matrix is both an upper triangular matrix and a lower triangular matrix. For example, 

M=[ 1 0 0 0 -2 0 0 0 -1 ]  is a diagonal matrix.

Identity matrix

An identity matrix is a diagonal matrix where the entries on the main diagonal are 1. An identity matrix is denoted by I. An identity matrix of size 2×2 is written as I2= [ 1 0 0 1 ] . An identity matrix of size 3×3 is written as I3= [ 1 0 0 0 1 0 0 0 1 ] . In general, the square matrix In is an identity matrix of size n×n.

Basic operations involving matrices

Transpose

Transpose of matrix 

M, denoted by MT is obtained by interchanging the rows and columns of  matrix M.

Given M= [ 1 0 -1 3 2 5 ] , the tranpose is MT= [ 1 -1 2 0 3 5 ] .

Scalar multiplication

To multiply a matrix M with a scalar k, each entry in the matrix is multiplied by k. 

For example,

      [ 21 2 3 4 5 6 ] = [ 2 4 6 8 10 12 ]  

      [ -51 0 1 3 -2 -1 ] = [ -5 0 -5 -15 10 5 ]  

Adding matrices

Matrix A and matrix B can be added only if they have the same size. The resulting matrix is obtained by adding each corresponding entries of the two matrices.

For example,

      [ 1 2 3 4 5 6 +2 -4 -6 2 1 2 ] = [ 1+2 2-4 3-6 4+2 5+1 6+2 ]  

                                                     = [ 3 -2 -3 6 6 8 ]

In the same way,

      [ 1 3 2 0 -2 3 0 0 -1 ] + [ 1 0 2 4 1 0 -4 5 1 ] = [ 2 3 4 4 -1 3 -4 5 0 ]

Subtracting matrices

As with addition, subtraction involving matrices can only be carried out if the matrices have same size. The resulting matrix is obtained by subtracting each corresponding entries of the two matrices.

For example,

     [ 1 2 3 4 5 6 ] - [ 2 -4 -6 2 1 2 ] = [ -1 6 9 2 4 4 ]

In the same way,

     [ 1 3 2 0 -2 3 0 0 -1 ] - [ 1 0 2 4 1 0 -4 5 1 ] = [ 0 3 0 -4 -3 3 4 -5 -2 ]