A matrix is an array of numbers arranged in rows and columns. In Mathematica matrices are expressed as a list of rows, each of which is a list itself. It means a matrix is a list of lists. If a matrix has *n* rows and *m* columns then we call it an *n* by *m* matrix. The value(s) in the ith row and jth column is called the *i, j* entry.

In mathematica, matrices can be entered with the { } notation, constructed from a formula or imported from a data file. There are also commands for creating diagonal matrices, constant matrices and other special matrix types.

## Creating matrices in Mathematica

- Create a matrix using { } notation

mat={{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}

but output will not be in matrix form, to get in matrix form use command like

mat//MatrixForm - Creating matrix using Table command

mat1=Table[b{row, column},

{row, 1, 4, 1}, {column, 1, 2, 1}]

];

MatrixForm[mat1] - Creating symbolic matrix such as

mat2=Table[x^{i}+x^{j }, {i, 1, 4}, {j, 1, 3}]

mat2//MatrixForm - Creating a diagonal matrix with nonzero entries at its diagonal

DiagonalMatrix[{1, 2, 3, r}]//MatrixForm - Creating a matrix with same entries i.e. a constant matrix

ConstantArray[3, {2, 4}]//MatrixForm - Creating an identity matrix of order
*n × n*IdentityMatrix[4]

## Matrix Operations in Mathematica

In mathematica matrix operations can be performed on both numeric and symbolic matrices.

- To find the determinant of a matrix

Det[mat] - To find the transpose of a matrix

Transpose[mat] - To find the inverse of a matrix for linear system

Inverse[mat] - To find the Trace of a matrix i.e. sum of diagonal elements in a matrix

Tr[mat] - To find Eigenvalues of a matrix

Eigenvalues[mat] - To find Eigenvector of a matrix

Eigenvector[mat] - To find both Eigenvalues and Eigenvectors together

Eigensystem[mat]

Note that +, *, ^ operators all automatically work element-wise.

## Displaying matrix and its elements

- mat[[1]] displays the first row of a matrix where mat is a matrix create above
- mat[[1, 2]] displays the element from first row and second column, i.e. m12 element of the matrix
- mat[[All, 2]] displays the 2nd column of matrix

References