# Instructions for Our Matrix Calculator

A complex matrix calculator is a matrix calculator that is also capable of performing matrix operations with matrices that any of their entries contains an imaginary number, or in general, a complex number. Such a matrix is called a complex matrix.

Our complex matrix calculator allows performing matrix algebra (addition, subtraction, multiplication, inverses, etc.) and calculate determinant, inverse, adjugate, rank, rref, and triangular forms of m × m matrices of any size with real or complex matrices and solve systems of linear equations with real or complex coefficients and real or complex column vectors.

You can increase the dimensions of a selected matrix by adding rows or columns to it by pressing the (insert row or insert column) buttons provided.

You can also conveniently set the number of rows or columns of a selected matrix by pressing the numbered buttons on the left of a row or above a column, respectively.

You can also set the dimensions of a selected matrix by entering the number of its rows or columns in the text boxes provided.

You can select and/or modify a matrix A, B, C, ..., H by first choosing them from the drop-down list on the right of the matrix calculator. These matrices are initially filled with 0's, except for the matrix A.

You can then set its dimensions and fill the matrix with numbers (real numbers, imaginary numbers or, in general, complex numbers) or expressions containing them.

This complex matrix calculator allows you to use any numeric (constant) expression, e.g., 1/2+3sin(5π/4)i for a matrix element.

This matrix calculator remembers the dimensions and entries of all matrices that you enter and also remembers whether a matrix is augmented for solving systems of linear equations.

All the data are retained indefinitely. To clear the data and reset the matrix calculator type in reset in the matrix expression box at the bottom of the matrix entries and press Clear, or clear site data from your browser history.

You can clear or fill a selected matrix by random numbers (integers between -10 and 10) by pressing the buttons provided. They do not internally replace the entries of the selected matrix but the random entries are used for calculations. You can retrieve the original entries that you entered manually by changing your device's orientation or refreshing the page.

## Matrix Algebra

To calculate the determinant, inverse, reduced row echelon form, adjugate, rank, lower/upper triangular forms and transpose of a selected matrix (A is initially selected) press the relevant buttons at the top of the matrix calculator.

You can do similarly as above with other matrices A, B, C, ..., H by first selecting them from the drop-down list and setting its dimensions and entries.

Under the Quick Calculations drop-down-list you can calculate frequently used matrix expressions involving two or more matrices such as A + B, (A+B)(C+D), and many more.

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

This matrix expression calculator allows you to use 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)

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.

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)

All 1 × 1 matrices are treated as scalars by this 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 1 × 1 matrices. For example, inv(2) is treated as inv([2]) which will be given as 0.5.

You can also use this 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 1×1 matrices. Assign numbers (or constant expressions) to the variables (i.e., 1×1 matrices) and press the Calculate button. The value of the function is given as a scalar.

## Solving Linear Systems

The system of linear equations solver 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). Note that 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 systems 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 additional 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 to be in the reduced row echelon form, then each system has a unique solution with the corresponding solution vector appearing on the right hand side.