Chapter 3- Determinants Flashcards Preview

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Flashcards in Chapter 3- Determinants Deck (11)
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1
Q

In the cofactor expansion, what is a and C?

A

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

2
Q

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

A

True

Theorem 2

3
Q

What are the row operations of determinants?

Let A be a square matrix

A

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

4
Q

What is the significance that

det A^T = det A

A

Column operations are equivalent to row operations when determining the determinant of A

5
Q

What is the multiplicative property for determinants?

A

det AB = (det A)(det B)

6
Q

True or false. Why?

det(A + B) = det A + det B

A

False.

This is not included in any of the theorems. Common mistake.

7
Q

True or false. Why?

If the columns of A are linearly dependent, then det A = 0

A

True.

If det A =/= 0, then A is an invertible matrix. Definition of such a matrix has columns that are linearly independent

8
Q

Let U be a square matrix such that

U^T*U = I. Show that detU = +1 or -1

A

Use theorem 6 and

det U^T = det U

9
Q

What is Cramer’s rule and how applicable is it?

A

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

10
Q

How to find the area of an ellipse using determinants?

A

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

11
Q

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)?

A

volume T(S) = |det A|*(volume S)