- #1
You would do better to evaluate this determinant using row and column operations. Start by subtracting row 2 from row 3, using calculations like $(n+2)^2 - (n+1)^2 = n^2+4n+4 - (n^2+2n+1) = 2n+3$: $$ \begin{vmatrix}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ (n+2)^2 & (n+3)^2 & (n+4)^2 \end{vmatrix} = \begin{vmatrix}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ 2n+3 & 2n+5 & 2n+7 \end{vmatrix}.$$ Then continue like this: $$\begin{aligned} \begin{vmatrix}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ (n+2)^2 & (n+3)^2 & (n+4)^2 \end{vmatrix} &= \begin{vmatrix}n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ 2n+3 & 2n+5 & 2n+7 \end{vmatrix} \\ \\ \text{(Subtract row 1 from row 2)}\qquad &= \begin{vmatrix}n^2 & (n+1)^2 & (n+2)^2 \\ 2n+1 & 2n+3 & 2n+5 \\ 2n+3 & 2n+5 & 2n+7 \end{vmatrix} \\ \\ \text{(Subtract col 2 from col 3)}\qquad &= \begin{vmatrix}n^2 & (n+1)^2 & 2n+3 \\ 2n+1 & 2n+3 & 2\\ 2n+3 & 2n+5 & 2 \end{vmatrix} \\ \\ \text{(Subtract col 1 from col 2)}\qquad &= \begin{vmatrix}n^2 & 2n+1 & 2n+3 \\ 2n+1 & 2& 2\\ 2n+3 & 2 & 2 \end{vmatrix} .\end{aligned}$$ Now subtract col 2 from col 3. Proceed in this way and you should end with a very simple answer, namely the constant $-8$.Monoxdifly said:
A determinant is a mathematical value that can be calculated for a square matrix. In the case of a matrix with polynomial elements, the determinant is calculated by using the coefficients of the polynomials as the entries of the matrix and then solving for the determinant using standard methods.
The determinant of a matrix with polynomial elements is calculated by using the coefficients of the polynomials as the entries of the matrix and then solving for the determinant using standard methods, such as the cofactor expansion method or the Gaussian elimination method.
The determinant of a matrix with polynomial elements is used in various mathematical applications, such as solving systems of equations, finding the inverse of a matrix, and determining if a matrix is invertible or singular.
Yes, the determinant of a matrix with polynomial elements can be negative. The determinant is a numerical value and can be positive, negative, or zero depending on the entries of the matrix.
Yes, there are several special properties of the determinant of a matrix with polynomial elements. For example, the determinant of a diagonal matrix is equal to the product of its diagonal entries, and the determinant of a triangular matrix is equal to the product of its diagonal entries. Additionally, the determinant of a matrix is equal to the product of its eigenvalues.
