Does det (AB) = det (A) det (B)?
This isn’t strictly an answer to the question because it is not a rigorous argument that det (AB) = det (A) det (B). But for me the idea I will share carries a lot of useful insight so I offer it in that spirit. It is based on the geometric interpretation of the determinant:
How do you find the diagonal form of det AB?
If A is singular, then AB is also singular, so det (AB)=0=det (A)det (B). For the nonsingular case we can row reduce A to diagonal form by Gauss-Jordan elimination (we avoid row-scaling). Every row-operation can be represented by an elementary matrix, the product of which we call E.
What are the terms in the inner sum of det (AB)?
So in the earlier equation for det (AB), the only terms in the inner sum that need be considered are those where ˉk defines a permutation τ. Show activity on this post. This isn’t strictly an answer to the question because it is not a rigorous argument that det (AB) = det (A) det (B).
How do you prove that a and B are nonsingular?
Assume that A and B are nonsingular, otherwise AB is singular, and the equation det (AB) = det (A) det (B) is easily verified. The key point is that we prove that the ratio d(A) = det (AB) / det (B) has the three properties.
What does the sign of Det (a) mean?
The sign of det (A) tells you whether A preserves or reverses orientation. Let n = 2 so we are dealing with areas in the plane. If A is a rotation matrix, then its effect on the plane is a rotation. det (A) is positive 1 because A actually preserves all areas (so absolute value 1) and preserves orientation (so positive).
What is the formula to find the value of det (AB)?
Ainjn) = ( det B)(εi1… inAi1j1… Ainjn) = ( det B)( det A)εj1… jn. Therefore det (AB) = det (A) det (B). Show activity on this post. P is permutation matrix, L lower triangular matrix, U upper triangular matrix.