Let k be a positive real number and let A = b | vedclass

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(A) First,calculate the determinant of matrix $A$. By performing row and column operations,we find $|A| = (2k+1)^3$.Since $B$ is a skew-symmetric matr

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$4$

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(A) First,calculate the determinant of matrix $A$. By performing row and column operations,we find $|A| = (2k+1)^3$.Since $B$ is a skew-symmetric matrix of order $3$,its determinant is $|B| = 0$.We know that $\det(\operatorname{adj} A) = |A|^{n-1}$,where $n$ is the order of the matrix. Here $n=3$,so $\det(\operatorname{adj} A) = |A|^2 = ((2k+1)^3)^2 = (2k+1)^6$.Similarly,$\det(\operatorname{adj} B) = |B|^2 = 0^2 = 0$.Given $\det(\operatorname{adj} A) + \det(\operatorname{adj} B) = 10^6$,we have $(2k+1)^6 = 10^6$.Taking the sixth root on both sides,$2k+1 = 10$.$2k = 9$,which implies $k = 4.5$.Therefore,$[k] = [4.5] = 4$.

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