Examples of solvings

• Compute the determinant:

  • a) by expanding along the i-th row;
  • b) by expanding down the j-th column;
  • c) first obtaining zeros in the i-th row.

  $\left|\begin{array}{cccc}1& 1& 3& 4\\ 2& 0& 0& 8\\ 3& 0& 0& 2\\ 4& 4& 7& 5\end{array}\right|,i=1,j=2;$

  For a solution you could use the page "Determinant calculator":

  • a) You should enter in the input field near the button "Expand along the row" the row number - 1. Then click this button. The solution will appear below.
  • b) You should enter in the input field near the button "Expand along the column" the column number - 2. Then click this button. The solution will appear below.
  • c) You should enter in the input field near the button "Get zeros in the row" the row number - 1. Then click this button. The solution will appear below.

• Evaluate a matrix expression to find the matrix K:

  $K=3AB-2CD,$

  $A=\left(\begin{array}{ccc}1& 2& 0\\ -1& -3& 4\\ 2& 5& -6\end{array}\right),B=\left(\begin{array}{cccc}1& 3& 6& -7\\ 2& 0& -1& -3\\ 0& 2& 4& 5\end{array}\right),C=\left(\begin{array}{cc}4& 3\\ -2& 0\\ 1& 4\end{array}\right),D=\left(\begin{array}{cccc}2& -1& 3& 0\\ 0& 2& 3& 5\end{array}\right);$

  To find the solution it is possible to use the page "Matrix calculator":

  1. Find on the page a button which adds matrix input tables, press it two times to get the input fields for matrices C and D.
  2. Enter the matrix A into the table "Matrix A", the matrix B into the table "Matrix B", the matrix C into the table "Matrix C", the matrix D into the table "Matrix D".
  3. Then enter the expression 3AB-2CD into the expression input field and press the button "=" next to the field.
  4. The result will appear below on the page.

• A problem. The enterprise lets out three kinds of production, using raw material of three types. Charges of each type of raw material by kinds of production and stocks of raw material at the enterprise are given in the table. To define volume of output of each kind at the set stocks of raw material.

  Type of raw material	Charge of raw material by kinds of production, weight/num.			Stock of raw material, weight units
  	1	2	3
  I	2	3	5	1030
  II	3	2	1	620
  III	1	1	3	510

  Let's make a system of the equations:

  2x_1 + 3x_2 + 5x_3 = 1030
3x_1 + 2x_2 + 1x_3 = 620
1x_1 + 1x_2 + 3x_3 = 510


  Let's use the page "System of equations calculator" to find the solution:

  1. Put the coefficients of the system into the input fields.
  2. Then press the button "Solve by Cramer's rule."