Gaussian elimination

The systems of linear equations:

can be solved using Gaussian elimination with the aid of the calculator.

In Gaussian elimination, the linear equation system is represented as an augmented matrix, i.e. the matrix containing the equation coefficients and constant terms with dimensions [n:n+1]:

Gaussian elimination

8 3 4 5 31 14 4 33 23 17 15 4 23 7 22 4 11 17 1 51

Linear equations system matrix

Calculation precision

Digits after the decimal point: 2

Calculate

Number of solutions

1

Solution vector

1.80

6.31

-1.55

0.77

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Details

Matrix	Row operations
8 3 4 5 31 14 4 33 23 17 15 4 23 7 22 4 11 17 1 51	- Multiply by 8, subtract 14 * row N 1 Multiply by 8, subtract 15 * row N 1 Multiply by 8, subtract 4 * row N 1
8 3 4 5 31 0 -10 208 114 -298 0 -13 124 -19 -289 0 76 120 -12 284	- - Multiply by -10, add 13 * row N 2 Multiply by -10, subtract 76 * row N 2
8 3 4 5 31 0 -10 208 114 -298 0 0 1464 1672 -984 0 0 -17008 -8544 19808	- - - Multiply by 1464, add 17008 * row N 3
8 3 4 5 31 0 -10 208 114 -298 0 0 1464 1672 -984 0 0 0 15928960 12263040	- - - Divide by 15928960
8 3 4 5 31 0 -10 208 114 -298 0 0 1464 1672 -984 0 0 0 1 0.77	Subtract 5 * row N 4 Subtract 114 * row N 4 Subtract 1672 * row N 4
8 3 4 0 27.15 0 -10 208 0 -385.76 0 0 1464 0 -2271.20 0 0 0 1 0.77	- - Divide by 1464
8 3 4 0 27.15 0 -10 208 0 -385.76 0 0 1 0 -1.55 0 0 0 1 0.77	Subtract 4 * row N 3 Subtract 208 * row N 3
8 3 0 0 33.36 0 -10 0 0 -63.08 0 0 1 0 -1.55 0 0 0 1 0.77	- Divide by -10
8 3 0 0 33.36 0 1 0 0 6.31 0 0 1 0 -1.55 0 0 0 1 0.77	Subtract 3 * row N 2
8 0 0 0 14.43 0 1 0 0 6.31 0 0 1 0 -1.55 0 0 0 1 0.77	Divide by 8
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Gaussian elimination

The method is named after Carl Friedrich Gauss, the genius German mathematician from 19 century. Gauss himself did not invent the method. The row reduction method was known to ancient Chinese mathematicians; it was described in The Nine Chapters on the Mathematical Art, a Chinese mathematics book published in the II century.

Forward elimination

The first step of Gaussian elimination is row echelon form matrix obtaining. The lower left part of this matrix contains only zeros, and all of the zero rows are below the non-zero rows:

The matrix is reduced to this form by the elementary row operations: swap two rows, multiply a row by a constant, add to one row a scalar multiple of another.
Our calculator gets the echelon form using sequential subtraction of upper rows , multiplied by from lower rows , multiplied by , where i - leading coefficient row (pivot row).
It is important to get a non-zero leading coefficient. If it becomes zero, the row gets swapped with a lower one with a non-zero coefficient in the same position.

Back substitution

During this stage the elementary row operations continue until the solution is found. Finally, it puts the matrix into reduced row echelon form:
,