PROPRIEDADES DOS DETERMINANTES

Properties involving determinants facilitate the calculation of their value in matrices that meet these conditions. 1st Property The value of the determinant of a matrix A is equal to the determinant of the matrix transposed to B, det A = det (A^t). 2nd Property When observing a matrix and verifying that the elements in a row or column are equal to zero, the value of its determinant will also be zero. If the elements are equal between two rows or two columns, the determinant of that matrix will be zero. If two rows or two columns have proportional elements in a matrix, the determinant will be zero. Observe the property between the 1st and 2nd rows. 3rd Property When we swap the positions of two rows or two columns in a matrix, the value of its determinant becomes the opposite of the determinant of the previous one. 4th Property When we multiply all the elements in a row or column of a matrix by a number K, its determinant is multiplied by K. The elements in the first row of P were multiplied by 2, so: det P' = 2 * det P CONSEQUENCE: If a square matrix A is multiplied by a real number k, its determinant is multiplied by k^n. det (k*A) = k^n * det A 5th Property A determinant is zero when a row is a linear combination of two other parallel rows. 8th Property The determinant of a triangular matrix is equal to the multiplication of the elements of the main diagonal. Remember that in a triangular matrix, the elements above or below the main diagonal are equal to zero. 9th Property Considering two square matrices of equal order and AB as the product matrix, we have: det (AB) = (det A) * (det B), according to Binet's theorem. 10th Property By multiplying all the elements in a row or column by the same number and adding the results to the corresponding elements in another row or column, we form the matrix B, where the following equality occurs: det A = det B. This theorem is attributed to Jacobi.