# How to Use The Online Matrix Calculator / Linear System Calculator

This online Matrix Calculator allows you to do matrix algebra (matrix addition, matrix multiplication, matrix inverse, etc.) and solve systems of linear equations with real or complex matrices.

To enter matrix entries first select a named matrix (A, B, C, ..., H) from the drop-down list and set its dimensions and fill the matrix — the matrix A is initially selected.

This Complex Matrix Calculator allows you to use any numeric (constant) expression, e.g., 1/2 + 2i + 3isin(3π/2) for matrix elements and the Complex Linear System Calculator allows augmented matrices with complex entries.

You can set the dimensions of a selected matrix in one of the following ways.

• By adding rows or columns to it by pressing the + (insert row or insert column) buttons provided.
• By entering the number of its rows or columns in the text boxes provided.
• By pressing the numbered buttons on the left of a row or above a column, respectively.

## Matrix Operations / Matrix Algebra

You can simply calculate determinant, inverse, rank Calculator, reduced row echelon form, adjugate, lower & upper triangular forms and transpose of a matreix by pressing the relevant buttons provided.

Under the Quick Calculations menu you can calculate frequently used matrix expressions involving up to eight matrices.

If a matrix expression is not listed under the quick calculation menu, you can enter it in the expression box provided and press the Calculate button.

This Matrix Expression Calculator allows use of any matrix expression which can be in the most general form, such as (2+sin(π/3))A + inv(A+B/det(A))(B/2 + BC^4)/D^(3+2^5)

You can use the following in your expressions: inv(), adj(), trans(), rref(), ut(), lt(), det()

1/A is the same as inv(A)

A/B is the same as Ainv(B)

If the matrix expression is a valid expression and contains no operations of incompatible matrices, the result will be displayed. Otherwise an error message is displayed.

All 1x1 matrices are treated as scalars by this online matrix calculator. They can be multiplied by any matrix (on either side) regardless of its dimension. Also if, for example, A = [1/2], then sin(A) is treated as sin(1/2). Conversely, whenever appropriate, scalars are treated as 1x1 matrices. For example, inv(2) is treated as inv([2]) which will be given as 0.5.

You can also use this online matrix calculator as a multi-variable function evaluator. Type in a function expression containing up to 8 variables (use A, B, C, ... as variables, instead of x,y,z, ...). Set all matrices involved as 1x1 matrices. Assign numbers (or constant expressions) to the variables (i.e., 1x1 matrices) and press the Calculate button. The value of the function is given as a scalar.

## Solving Linear Systems

The system of linear equations calculator allows two different ways to solve such systems.

With the first method, which is a neat way to solve a single linear system, select the Linear System checkbox provided. Set the dimensions of the coefficient matrix and fill the augmented matrix with real or complex numbers (or expression of them) noting the last column highlights the rhs (right hand side) of the system. Now press the Solve button. If the system is consistent and has a unique solution, the vector representing the unique solution will be displayed together with the RREF (reduced row echelon form) of the augmented matrix. If there are more than one solutions, the general solution is given. If the system is inconsistent a message stating so and the RREF will be displayed.

With the second method you can solve more than one linear system at once, all having the same square matrix m × m as their coefficient matrix. Do not select Linear System but set the number of columns more than the number of rows. Fill each column after the mth column with the right hand side of the corresponding system (with the same coefficient matrix) and press the RREF button. If you see the entries of the m × m matrix on the left of the reduced row echelon form, then each system has a unique solution with the corresponding solution vector appearing on the right hand side.