TEST (Ch 2 - 3)

Question 1

Matrix is invertible iff det(A) ≠ ?

Answer 1

det(A) ≠ 0

Question 2

Inverse of a (2x2) matrix:

Answer 2

Question 3

The determinant can only be calculated for what kind of matrices?

Answer 3

SQUARE matrices (nxn)

Question 4

What is the det of a (1x1) matrix?

Answer 4

det(A) = A

Question 5

What is the determinant of a (2x2) matrix?

Answer 5

det(A) = ad - bc

Question 6

What is the minor of entry a_ij (M_ij)?

Answer 6

the det of submatrix that remains after the ith row and the jth column are deleted from A.

Question 7

What is the cofactor of entry a_ij (C_ij)?

Answer 7

Question 8

What is the relationship between a minor and its corresponding cofactor?

Answer 8

They are either the same or negatives of each other

Question 9

What is cofactor expansion?

Answer 9

obtained by multiplying the entries in any row/column by the cofactors and adding the resulting product = determinant

Question 10

Does it matter which row/column you choose when doing cofactor expansion?

Answer 10

Get the same determinant no matter the row/column you choose

Easiest: choose the one with the most 0s

Question 11

What is the determinant of a triangular matrix?

Answer 11

det(A) = product of entries on the main diagonal

Question 12

What is a simple technique for evaluating the det of a (2x2) or (3x3) matrix?

Answer 12

Question 13

If a matrix has a row or column of 0s, what is its det?

Answer 13

det(A) = 0

Question 14

If a matrix has a 2 proportional rows/columns, what is its det?

Answer 14

det(A) = 0

Question 15

What is the det(A^T)?

Answer 15

det(A^T) = det(A)

Question 16

Elementary Row Operations on Determinants

If single row/column was multiplied by k, what is det?

Answer 16

det(B) = k det(A)

Question 17

Elementary Row Operations on Determinants

If single two rows/columns were interchanged, what is det?

Answer 17

det(B) = –det(A)

Question 18

Elementary Row Operations on Determinants

If a multiple of a row/column was added to another, what is det?

Answer 18

det(B) = det(A)

Question 19

How can you evaluate the det using Row Reduction?

Answer 19

1. Reduce matrix to REF (upper triangular)
2. Cofactor expansion or main diagonal

Question 20

Is there an addition formula for determinants?

Answer 20

NO!

det(A+B) ≠ det(A) + det(B)

Question 20

What is the scalar multiplication formula for determinants?

Answer 20

det(kA) = kⁿ det(A)

Question 21

What does invertible mean?

Answer 21

it can be expressed as a product of elementary matrices

Question 22

If A and B are square matrices of the same size:

det(AB) =

Answer 22

det(AB) = det(A) det(B)

Question 23

What is the det(A^-1)?

Answer 23

Question 24

What is an **adjoint**?

Answer 24

Transpose of the matrix of cofactors of A

Question 25

What is **Cramer's Rule**?

Answer 25

* helps solve for UNIQUE solution of a system * First double-check that det(A) ≠ 0

Question 26

Vectors are equivalent if \_\_\_

Answer 26

IF they have the same magnitude and direction Or if all of their respective components are equal

Question 27

What is a **zero vector**?

Answer 27

Initial & terminal points coincide has length 0 Has no natural direction, it can be assigned to any direction that’s convenient

Question 28

What is the parallelogram rule (+ and – vectors)?

Answer 28

Question 29

What is the difference between parallel & collinear with vectors?

Answer 29

Parallel = collinear for vectors → because parallel vectors can be moved to become collinear

Question 30

Component form of vectors

Answer 30

Question 31

What is the formula for a vector whose initial point is NOT the origin?

Answer 31

Question 32

**N-space** (Rⁿ)

Answer 32

set of ordered n-tuples

Question 33

Algebraic properties of vectors:

Answer 33

prove using components

Question 34

What is a **linear combination**?

Answer 34

The ks are the coefficients of the linear combination

Question 35

What is a **norm** (|| ||) of a vector?

Answer 35

length/magnitude of vector

Question 36

Properties of norms

Answer 36

Question 37

What is a **unit vector**?

Answer 37

vector magnitude = 1

Question 38

**Standard unit vector**

Answer 38

v = (v₁, v₂, v₃ … v_n)

Question 39

What is the distance between two vectors?

Answer 39

Question 40

**Dot product** definition

Answer 40

Product of magnitude of component **v** in the direction of **w** with the magnitude of **w**.

Question 41

If two vectors are oriented very similarly then there dot product is \_\_

Answer 41

The more similarly two vectors are oriented the higher the dot product

Question 42

**Geometric** definition of dot product

Answer 42

Question 43

What does the dot product tell us about the angle between two vectors?

Answer 43

Question 44

**Algebraic** definition of the dot product

Answer 44

Question 45

Expressing the norm of a vector with dot product

Answer 45

Question 46

Properties of the dot product

Answer 46

Question 47

Cauchy-Shwarz Inequality

Answer 47

Question 48

Answer 48

like a triangle (no side can be longer than the sum of the other sides)

Question 49

Two vectors are orthogonal if \_\_

Answer 49

dot product = 0

Question 50

Are 0 vectors orthogonal to anything?

Answer 50

Zero vectors in Rⁿ are perpendicular to every vector in Rⁿ

Question 51

What is a **normal**?

Answer 51

vector orthogonal to line/plane

Question 52

**Normal**: Point-normal form

Answer 52

Question 53

**Normal**: Component form

Answer 53

Through point P = (x₀, y₀, z₀), **n** = (a, b, c)

Question 54

Vector form of a line/plane through the origin

Answer 54

Question 55

*ax + by + c = 0* represents what?

Answer 55

a line in R² with normal **n** = (a, b)

Question 56

*ax + by + cz + d = 0* represents what?

Answer 56

a plane in R³ with normal **n** = (a, b, c)

Question 57

What is an **orthogonal projection**?

Answer 57

Question 58

Vector component of **u** along **a**

Answer 58

Question 59

Vector component of **u** orthogonal **a**

Answer 59

Question 60

Norm (|| ||) of a vector projection

Answer 60