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TYPES primer

How to find the determinant of the matrix

We find the determinant of a matrix of four rows and four columns

The determinant of the matrix is ​​a polynomial of all possible products of its elements.

One way to calculate the determinant of a matrix decomposition of the column for additional minors (submatrix).

You will need

  • - a pen
  • - paper



It is known that the determinant of the secondthe order is calculated as follows: the product of the elements of the main diagonal elements of the subtracted product of the secondary diagonal. Therefore it is convenient to decompose the matrix on minors of the second order, and then calculate the determinants of these minors, as well as the determinant of the original matrix.
The figure shows the formula for calculating thethe determinant of any matrix. Using it, first we expand the matrix of the third order of the minors, and then each received minor on the minors of the second order, which makes it easy to calculate the matrix determinant.

We use this formula for the decomposition of the original matrix by the first column


We expand on the formula of the original matrixAdditional size 3 by 3 matrix of additional matrix, or minor, are formed by deleting from the initial matrix of one row and one column. The number of such minors include polynomials multiplied by the element of the matrix to which they are added, a sign of the polynomial is determined by the degree of -1, which is the sum of the element indices.

Degradation of the matrix to the minors of third order


Now, each of the matrices of the third orderdecompose in the same way in the second-order matrix. Find the determinant of such a matrix and each obtain a number of polynomials of the elements of the original matrix, go on a purely arithmetic.

Decomposition to minors of the second order and the calculation of the determinant of the matrix

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