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Gaussian elimination

The calculator solves the systems of linear equations using row reduction (Gaussian elimination) algorithm. The calculator produces step by step solution description.

The systems of linear equations:
\begin{cases}a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1\\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2\\ \dots \\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m\\ \end{cases}
could be solved using Gaussian elimination with aid of our calculator.

In Gaussian elimination, the linear equation system is represented as an augmented matrix, i.e. the matrix containing the equation coefficients a_{ij} and constant terms b_i with dimensions [n:n+1]:
\begin{array}{|cccc|c|}  a_{11} &  a_{12} &  ... &  a_{1n} &  b_1\\  a_{21} &  a_{22} &  ... &  a_{2n} &  b_2\\  ... &  ... &  ... &  ... &  ...\\  a_{n1} &  a_{n2} &  ... &  a_{nn} &  b_n\\ \end{array}

PLANETCALC, Gaussian elimination

Gaussian elimination

Digits after the decimal point: 2
Number of solutions
 
Solution vector
 

Gaussian elimination

The method is named after Carl Friedrich Gauss, the genious German mathematician of 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, Chinese mathematics book, issued in 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:
\begin{array}{|cccc|c|}  a_{11} &  a_{12} &  ... &  a_{1n} &  \beta_1\\  0 &  a_{22}  &  ... &  a_{2n} &  \beta_2 \\ 0 & 0 & \ddots & \vdots & \vdots \\ 0 &  0 &  0 & a_{nn} &  \beta_n\\ \end{array}

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 A_i, multiplied by {a_{ji}} from lower rows A_j , multiplied by {a_{ii}}, where i - leading coefficient row (pivot row).
It is impotant to get non-zero leading coefficient. If it becomes zero, the row get swapped with lower one with 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:
\begin{array}{|cccc|c|}  1 &  0 &  ... &  0 &  \beta_1\\  0 &  1 &  \vdots &  0 &  \beta_2 \\ 0 & 0 & \ddots & \vdots & \vdots \\ 0 &  0 &  0 & 1 &  \beta_n\\ \end{array} ,
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