Chapter 3- Determinants Flashcards
In the cofactor expansion, what is a and C?
a is the entry in the i row and j column. a alternates in plus and minus signs, depending on position.
C is the determinant of the submatrix from removing the i row and j column
Cofactor expansion of first row
det A = a11C11 + … + a1nC1n
True or false. Why?
If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A
True
Theorem 2
What are the row operations of determinants?
Let A be a square matrix
a) If a multiple of one row of A is added to another row to produce B, then det B = det A
b) If two rows of A are interchanged to produce B, then
det B = - det A
c) If one row of A is multiplied by k to produce B, then
det B = k * det A
What is the significance that
det A^T = det A
Column operations are equivalent to row operations when determining the determinant of A
What is the multiplicative property for determinants?
det AB = (det A)(det B)
True or false. Why?
det(A + B) = det A + det B
False.
This is not included in any of the theorems. Common mistake.
True or false. Why?
If the columns of A are linearly dependent, then det A = 0
True.
If det A =/= 0, then A is an invertible matrix. Definition of such a matrix has columns that are linearly independent
Let U be a square matrix such that
U^T*U = I. Show that detU = +1 or -1
Use theorem 6 and
det U^T = det U
What is Cramer’s rule and how applicable is it?
Helps solve Ax = b (A is invertible)
The unique solution x has ENTRIES
xi = |det Ai b| / (det A)
i = 1, 2, … , n
Where Ai b is the matrix A replacing the ith column with b
Incredibly inefficient for matrices larger than 2x2
How to find the area of an ellipse using determinants?
Let T map R^2 to R^2 be the linear transformation by a 2x2 matrix
area of ellipse = |det A|*(area of circle)
Where A can transform circle to ellipse
Check 3.3 example 5 for reinforcement
If T is determined by a 3x3 matrix, and S is a closed shape in R^3, then what is the volume of T(S)?
volume T(S) = |det A|*(volume S)
