min asked in Science & MathematicsMathematics · 6 months ago

How is the determinant of a determinant equal to the determinant of the matrix to the power of its number of rows?

How is the following true?:

det((det(A))I) = (det(A))^n(det(I))=(det(A))^n

I understand that det((det(A))I) = det(det(A))det(I) and that the determinant of the Identity matrix is 1, but don't understand the relation that det(det(A))=(det(A))^n.

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  • JOHN
    Lv 7
    6 months ago
    Favorite Answer

    det(A)I is a diagonal matrix with elements in the leading diagonal each of value det(A). And the determinant of a diagonal matrix is the product of the diagonal elements. This is how you get (detA))^n. Also note that det (I) = 1. The whole line det((det(A))I) = (det(A))^n(det(I))=(det(A))^n follows from this.

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    • JOHN
      Lv 7
      6 months agoReport

      Thus you can't say det(det(A)A) = det(det(A))det(A). In particular det(A)I is not a matrix product - is not the product of a 1 x 1 matrix and an n x n matrix (n ≠ 1).

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