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Determinant:
Enter a determinant, with each element separated by a comma and each line ending with a semicolon.
Note that mathematical functions and variables are not supported.
Enter a determinant, with each element separated by a comma and each line ending with a semicolon.
Note that mathematical functions and variables are not supported.
Current location：Linear algebra >Determinant >History of determinant >Answer
$$ \begin{aligned} \color{black}{Calcu}&\color{black}{late\ determinant\ \ \begin{vmatrix} &1 &1 &1 &1\\ & 1 &2 &-1 &4\\ & 2 &-3 &-1 &-5\\ & 3 &1 &2 &11\\\end{vmatrix} \ \ .}\\Solut&ion：\\&\begin{vmatrix} &1 &1 &1 &1\\ &1 &2 &-1 &4\\ &2 &-3 &-1 &-5\\ &3 &1 &2 &11\\\end{vmatrix}\\\\ &\color{grey}{The\ ratio\ of\ elements\ in\ the\ second\ and\ first\ rows\ of\ the\ first\ column\ is:\ 的比值为\ 1\ ,line\ 2\ -\ \ 1\ ×\ line\ 1\ } \\\\\ =\ &\begin{vmatrix} &1 &1 &1 &1\\ &0 &1 &-2 &3\\ &2 &-3 &-1 &-5\\ &3 &1 &2 &11\\\end{vmatrix}\\\\ &\color{grey}{The\ ratio\ of\ elements\ in\ the\ third\ and\ first\ rows\ of\ the\ first\ column\ is:\ 的比值为\ 2\ ,line\ 3\ -\ \ 2\ ×\ line\ 1\ } \\\\\ =\ &\begin{vmatrix} &1 &1 &1 &1\\ &0 &1 &-2 &3\\ &0 &-5 &-3 &-7\\ &3 &1 &2 &11\\\end{vmatrix}\\\\ &\color{grey}{The\ ratio\ of\ elements\ in\ the\ 4th\ and\ first\ rows\ of\ the\ first\ column\ is:\ 的比值为\ 3\ ,line\ 4\ -\ \ 3\ ×\ line\ 1\ } \\\\\ =\ &\begin{vmatrix} &1 &1 &1 &1\\ &0 &1 &-2 &3\\ &0 &-5 &-3 &-7\\ &0 &-2 &-1 &8\\\end{vmatrix}\\\\ &\color{grey}{The\ ratio\ of\ elements\ in\ the\ third\ and\ second\ rows\ of\ the\ second\ column\ is:\ 的比值为\ -5\ ,line\ 3\ line\ +\ 5\ ×\ line\ 2\ } \\\\\ =\ &\begin{vmatrix} &1 &1 &1 &1\\ &0 &1 &-2 &3\\ &0 &0 &-13 &8\\ &0 &-2 &-1 &8\\\end{vmatrix}\\\\ &\color{grey}{The\ ratio\ of\ elements\ in\ the\ 4th\ and\ second\ rows\ of\ the\ second\ column\ is:\ 的比值为\ -2\ ,line\ 4\ line\ +\ 2\ ×\ line\ 2\ } \\\\\ =\ &\begin{vmatrix} &1 &1 &1 &1\\ &0 &1 &-2 &3\\ &0 &0 &-13 &8\\ &0 &0 &-5 &14\\\end{vmatrix}\\\\ &\color{grey}{The\ ratio\ of\ elements\ in\ the\ 4th\ and\ third\ rows\ of\ the\ third\ column\ is:\ 的比值为\ \frac{5}{13}\ ,line\ 4\ -\ \frac{5}{13}\ ×\ line\ 3\ } \\\\\ =\ &\begin{vmatrix} &1 &1 &1 &1\\ &0 &1 &-2 &3\\ &0 &0 &-13 &8\\ &0 &0 &0 &\frac{142}{13}\\\end{vmatrix}\\\\&\color{grey}{It\ has\ been\ transformed\ into\ an\ upper\ triangular\ determinant.}\\&\color{grey}{Multiply\ the\ elements\ of\ the\ main\ diagonal\ of\ the\ upper\ triangular\ determinant\ to\ obtain\ the\ result.} \\\\ \ =\ &-142\end{aligned}$$
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The main properties of determinant:
Property 1 . The rows and columns of a determinant are swapped in order, and their values remain unchanged.
Property 2 . If the positions of two rows (or columns) of a determinant are swapped, the determinant only changes its sign.
Property 3 . If the elements of a row (column) in a determinant have a common factor K, then K can be mentioned outside the sign of the determinant.
Property 4 . 若If the elements of a certain row in a determinant can be expressed as the sum of two terms, then this determinant is equal to the sum of these two determinants. These two determinants remain unchanged except for the element in this row, which is one of the two terms.
Property 5 . (i) The elements of a certain row (column) in the determinant are all 0; (ii) The two rows (columns) of the determinant are exactly the same; (iii) The elements of the two rows (columns) of the determinant are proportional. If one of the above conditions is true, the value of the determinant is zero.
Property 6 . If a row (column) of a determinant is changed λ If multiplied to another row (column), the value of the determinant remains unchanged.