Determinant of diagonal matrix?

Use induction (this is a must) to prove that the determinant of an nxn matrix is equal to the product of it’s diagonal.

4 Answers

  • *Assuming that the matrix is diagonal…

    ———————-

    Base Case:

    This assertion is certainly true for a 1 x 1 matrix (essentially a scalar).

    Inductive Step:

    Suppose that this claim is true for any k x k diagonal matrix.

    To show that this is true for a (k+1) x (k+1) diagonal matrix (call it A),

    first expand across the first row.

    So, the determinant of A equals a(1,1) * [determinant of a k x k diagonal matrix],

    since all other entries on the first row of A are 0’s.

    However, we know by inductive hypothesis that the determinant of a k x k matrix

    is the product of its diagonal entries, which in this case are a(2, 2), a(3, 3), …, a(k, k).

    Hence, |A| = a(1, 1) * [a(2, 2) * a(3, 3) * … * a(k, k)] = product of its diagonal entries.

    This completes the induction.

    I hope this helps!

  • Determinant Of Diagonal Matrix

  • RE:

    Determinant of diagonal matrix?

    Use induction (this is a must) to prove that the determinant of an nxn matrix is equal to the product of it’s diagonal.

  • https://shorturl.im/A0lkU

    If you’re trying to extend the little trick people use with 3x3s, where you write the first columns after the last column and add or subtract diagonal product, it doesn’t work. That technique doesn’t extend to matrices beyond 3×3. It doesn’t work for 2x2s either. Most mathematics instructors don’t teach it because students get confused thinking it should be a general technique.

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