Chapter 1 - Determinants Flashcards
Det A
ad-bc
A^-1
If detA isn’t 0, then
A^-1 = 1/(ad-bc) * matrix of [d -b]
[-c a]
What statements are equivalent to detA not equal to 0?
The columns of A are linearly independent.
The rows of A are linearly independent.
The linear map La:R^2 to R^2 is a bijection
Cross product of u=(u1,u2,u3) and v=(v1,v2,v3)
u × v = (u2v3-u3v2, u3v1-u1v3, u1v2-u2v1)
i × j j × i j × k k × j k × i i × k
= k = -k = i = -i = j = -j
-u × v
u × u
= u × v
= 0
u × v|
= |u| |v| sin(theta)
where theta is the angle between u and v, taken such that its between 0 and pi
Volume of skewbrick with sides u,v,w
The absolute value of u • (v × w) i.e determinant of the matrix: [u1 u2 u3] [v1 v2 v3] [w1 w2 w3]
Are u × (v × w) and (u × v) × w equal?
No
Are u • (v × w), v • (w × u), and w • (u × v) equal?
Yes
Det (AB)
= det(A) * det(B)
Linearly dependent vectors
Let r1,r2,r3 be vectors in R^3.
They are linearly dependent if there are scalars (real numbers) t1, t2, t3, not all 0, such that
t1r1 + t2r2 + t3r3 = 0
Linearly independent vectors
If the only scalars t1, t2, t3 such that
t1r1 + t2r2 + t3r3 = 0,
are t1 =0, t2=0, t3=0
Or if r1 × r2 doesn’t equal 0
Or if r1 • (r2 × r3) doesn’t equal 0
What statements are equal to the determinant of a 3x3 matrix not equalling 0?
The columns of A are linearly independent.
The linear map La:R^3 to R^3 is injective
Basic properties of determinants: Det (r2,r1) Det (r1+r1', r2) Det(tr1, r2) Det (i,j)
Det (r2,r1)= -det (r1,r2)
Det (r1 + r1’, r2) = det (r1, r2) + det (r1’, r2)
Det (tr1, r2) = tdet (r1, r2)
Det (i, j) = 1
