The determinant or determinant of the matrix - is a number calculated by a special formula made up of combinations of its members.
Just say that the determinant can be calculated only for a square matrix.
The determinant of the matrix will be calculated as followsmanner. This will be the sum of the coefficients of the first row, each of which multiply by the determinant of the matrix obtained by deleting from the original column and row in which the ratio is multiplied. Signs of these factors will rotate (in the first is "+", the second is "-", etc.).
Note that this formula is true for all elements of the rows - not necessarily take first, it's just more convenient for clarity.
There is a second method. There is a certain algorithm for computing.
First, we introduce the concept of the main diagonal of the matrix - the elements that stand diagonally, starting from and ending a11 and (nn) (that is, from top left to bottom right).
So back to the algorithm.
For one matrix element of the determinant is equal to the value of the element.
For the 2x2 matrix it will be the difference between the products of the elements, standing on the main and secondary diagonal (similar, side diagonal goes from the upper right to the lower left corner).
For the 3x3 matrix it will do so: the first two columns to the right of the third sign again. It turns out, as it were 3x5 matrix. It is as if it's just the reception. Further works are summed elements to receive three main diagonals and three side. These amounts are deducted. The resulting number will be the determinant of the matrix.
The picture shows another embodiment of the method of calculating the same, then just do without appending, just multiply the elements and subtract the sum of the products according to the scheme.
For a matrix of 4x4, 5x5, etc. It is generally the same will be implemented, but here there are difficulties due to the large amount of numbers and multiplications / additions to be executed, so that the risk of making a mistake. Therefore, in such cases, advantageous to use the first method.
Note that the determinant of the identity matrix is one, as is easily seen.