Matrices And Determinants

To find the determinant of , we start by considering the properties of the matrix .Given the matrix : The determinant of is computed as: Since the determinant of is 1, for any power : Thus, regardless of the power , the determinant of remains 1.Now, for the expression , we note the alternating signs. The determinant of can be computed by the linearity property of determinants, combining each term's determinant: Substituting the known determinants (all equal to 1): Since there are 2020 terms and they alternate between 1 and -1, if you sum these terms, every pair of terms (consisting of a +1 and a -1) cancels out, resulting in: Thus, the determinant of is 1.

A skew-symmetric matrix is defined as a square matrix that satisfies the condition . For a skew-symmetric matrix, this implies:$ a_{11}, a_{22} a_{33} a_{12}, a_{13} a_{23} \{-3,-2,-1,0,1,2,3\} a_{12}, a_{13}, a_{23} $

Given the matrix , we will determine the determinant of for any natural number .The determinant of matrix can be calculated as follows: Simplifying each term, we get: Using the Pythagorean identity, we know that .Therefore, .Since the determinant of a product of matrices is the product of their determinants, we have: Thus, for any natural number , the determinant of is 1.

Firstly,$ $$ $$ C_2 \rightarrow C_2-C_1-C_3 C_2 $

Step 1. Compute the determinant Simplify: So the condition becomes: Step 2. Solve the inequalitiesLeft side: Add 1 to both sides: Divide by 15: Right side: Add 1 to both sides: Divide by 15: Step 3. Combine Thus, ✅Correct Option: A.