Matrix equations calculator

* X =

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Solution comments
Without description (answer only)

a

b

c

d

x

y

z

clear

i

ab
x2
xn

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3131313131351515151515≈83137
2
2510
=Solve

  About matrix equations calculator

This is a free online matrix equations calculator with complete, detailed, step-by-step description of solutions, that performs operations with matrices up to 99x99 in size with matrix elements of this type: decimal numbers, fractions, complex numbers, variables.

To start the calculation, you need to first enter the size of the matrix in the input field that you can find from the very top of the screen, also there you can choose the desired method of calculation.

A little below you will find a matrix window in which you need to enter matrix elements using the keyboard. The matrix control panel is also located here, which simplifies work with matrices and contains the following control elements:

  • The first element allows you to expand the matrix window. This can be especially useful in cases where you need to perform calculations with very large matrices that do not fit completely. If the matrix is still not visible after expanding the window, you can change the scale of the matrix using the + / - buttons;
  • The second element performs the function of copying the matrix input to the memory buffer. This can be useful in cases where you often use the same matrix for calculations, or if you need to move matrices between operations;
  • And the last element inserts the previously copied matrix, which allows you to speed up the process of entering the matrix to just a few clicks, instead of doing it manually;

And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with it. It is visually divided into three parts, each of which is responsible for the following functionality:

  • The first allows you to select the number format when the solution result is displayed. Also, here you can turn off comments to the solution of the problem if you have already understood how to solve this problem, and you use the calculator to speed up or check your own calculations. Or you can turn off the step-by-step solution entirely if you only need the result of the solution;
  • The second contains buttons that allow you to change the type of the matrix input field, erase its elements or the entire matrix, and the largest button with an equal sign, which will take you to the screen with the solution of the problem. All these buttons are duplicated by keys on the keyboard. To find out which key on the keyboard to press, simply hover over one of the buttons and a tooltip will appear with the name of the key. You can also use the arrow keys on your keyboard to move the cursor between matrix input fields;
  • And the last one allows you to choose the number of digits after the decimal point for rounding non-integer numbers. Also, here you can immediately see an example of how rounded fractions will look;

  What are matrix equations?

Matrix equations are an algebraic expression in which a matrix with known coefficients and a matrix of unknown variables are located on the left side of the equation, and a matrix of constants is located on the right side of the equation, which is the result of multiplying a matrix with known coefficients by a matrix of unknown variables. The essence of matrix equations is to find a matrix with unknown variables.

  How to calculate matrix equations?

To solve a matrix equation, we need to find the inverse matrix of the matrix with known coefficients and multiply it by the matrix of constants, and this will give us the matrix of unknown variables, which is the result of the matrix equation.

  Example of calculating matrix equations

Write the initial matrix
A
:
A
=
71
7
2
4
8
8
5
5
5
5
8
5
2
2
7
2
Write the initial matrix
B
:
B
=
7
7
2
4
8
8
5
5
5
5
8
5
2
2
7
2
We have a following matrix equation format:
A
*
X
=
B
;
To find the roots of the equation, we can get rid of the matrix
A
on the left side of the equation;
To do this we can multiply the left and right parts of the equation by
A
-1
0
on the left;
After that we get the following equation:
A
-1
0
*
A
*
X
=
A
-1
0
*
B
;
We can simplify this equation;
Any matrix multiplied by the inverse matrix is equal to an identity matrix;
Any matrix multiplied by an identity matrix is equal to the same matrix;
Knowing this we have the following:
A
-1
0
*
A
*
X
=
A
-1
0
*
B
; =>
E
*
X
=
A
-1
0
*
B
; =>
X
=
A
-1
0
*
B
;
To find the roots of the equation, we calculate the inverse matrix of matrix
A
and multiply it by matrix
B
on the left;
2
Matrix inverse A⁻¹
A
-1
0
=
1
50
-
1
50
0
1
100
-
1
50
7
20
-
11
25
13
50
0
0
-
11
100
13
50
0
-
33
100
81
100
-
17
25
3
X = A⁻¹ * B
X
=
A
-1
0
·
B
=
1
50
-
1
50
0
1
100
-
1
50
7
20
-
11
25
13
50
0
0
-
11
100
13
50
0
-
33
100
81
100
-
17
25
·
7
7
2
4
8
8
5
5
5
5
8
5
2
2
7
2
=
0
1
-
1
20
-
37
100
0
1
0
0
0
0
1
0
0
0
0
1
Answer
A · X = B
0
1
-
1
20
-
37
100
0
1
0
0
0
0
1
0
0
0
0
1
SIZE4×4FORMA · X = B

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