5.6 The determinant and trace

Ah, the determinant, a function on square matrices that produces values in F. It’s so elegant, so useful, tells us so much, and is such an annoying and error-prone thing to compute beyond the 2-by-2 case. matrix$determinant determinant

Let’s look at its properties before we discuss its calculation. Let A be an n-by-n matrix. We denote its determinant by det(A):

• det(A) ≠ 0 if and only if A is invertible.
• For b a scalar in F, det(bA) = bⁿ det(A).
• If any row or column of A is all zeros, then det(A) = 0. The determinant being zero does not imply a row or column is zero.
• If A is upper or lower triangular, the determinant is the product of the diagonal entries. If one of those diagonal entries is 0, the determinant is thus 0.
• In particular, det(I) = 1 for I an identity matrix.
• The determinant behaves well when taking transposes and conjugates:

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