Find the Determinant of the Resulting Matrix

Problem

det([[2,5,2,2],[2,5,2,2],[1,21,2,2],[1,1,1,1]]⋅[[5,7,7,10],[3,3,2,1],[3,47,58,5],[4,4,10,10]])

Solution

1. Apply the determinant property for the product of two matrices, which states that det(A*B)=det(A)*det(B)

2. Identify the first matrix A and observe its rows.

A=[[2,5,2,2],[2,5,2,2],[1,21,2,2],[1,1,1,1]]

3. Check for linear dependence in matrix A Notice that the first row and the second row are identical.

(Row_1)=[2,5,2,2]

(Row_2)=[2,5,2,2]

4. Determine the determinant of matrix A Since a matrix with two identical rows has a determinant of zero, we find:

det(A)=0

5. Calculate the final product using the property from step 1.

det(A*B)=0⋅det(B)

det(A*B)=0

Final Answer

det([[2,5,2,2],[2,5,2,2],[1,21,2,2],[1,1,1,1]]⋅[[5,7,7,10],[3,3,2,1],[3,47,58,5],[4,4,10,10]])=0

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