Matrices

Question 1

The range of a matrix is

Answer 1

the order of the largest square sub-matrix whose determinant is not 0

Question 2

The range of A is written as

Answer 2

Rag A or rg(A).

Question 3

The order of a square matrix is the

Answer 3

number of rows or columns in that matrix.

Question 4

An array of order 2 is the same as saying ____________, which has ______ and _______.

Answer 4

an array of dimensions 2×2
2 rows
2 columns

Question 5

A submatrix is

Answer 5

a matrix that is contained within another matrix

Question 6

Within a matrix, we can choose rows and columns that form another independent matrix.

Answer 6

True

Question 7

I can choose columns that are not sequential to form a submatrix.

Answer 7

True

Question 8

I can choose fewer rows and columns to form a submatrix.

Answer 8

True

Question 9

To calculate the range of a matrix, we must ____________.

Answer 9

choose the sub-matrix with the highest possible order and calculate its determinant

Question 10

The _______ of the largest SQUARE sub-matrix, whose determinant is ________, will be the range of the matrix.

Answer 10

order

not zero

Question 11

___________ matches the order of the largest square sub-matrix we have found, whose determinant is other than zero.

Answer 11

The range

Question 12

Linearly independent vectors are ______________, i.e. which cannot be composed from the linear composition of the rest of the vectors.

Answer 12

the vectors whose formation does not depend on any other vector in the matrix

Question 13

The linear vectors of an array is ______________.

Answer 13

the number of vectors other than zero that remain after having triangulated the array formed by them

Question 14

The vectors of an array we are referring to are the _________, i.e. each row of the array corresponds to a vector.

Answer 14

row vectors

Question 15

To obtain the number of linearly independent vectors, we are going to ____________, that is to say, we are going to make the ___________ zeros.

Answer 15

triangulate the matrix

elements below the main diagonal

Question 16

The first element of the first column needs to be a 1, if we are going to triangulate the matrix.

Answer 16

True

Question 17

If we have a 3x4 matrix with a row full of zeros, this matrix has ____________ vectors and as a consequence, the other vector is dependent on the other two.

Answer 17

two linearly independent

Question 18

If the first element of a matrix is not a 1, we can exchange it for a vector in matrix that has 1 as its element.

Answer 18

True

Question 19

When calculating the range from the Gaussian Method, if we have already triangulated a 3x4 matrix, and we are not left with any row with zeros, then the 3 vectors of the matrix are ___________. Therefore, as the matrix has 3 linearly independent vectors, its range is ___.

Answer 19

linearly independent

3

Question 20

When triangulating a 3x3 matrix and we have 2 zeros in the first column and 1 zero in the second column, since the matrices had 3 rows, remember that to triangulate the matrix is to make ___________ and that in each case it will be different, depending on the number of rows in the matrix.

Answer 20

zeros below the main diagonal

Question 21

In the Gaussian Method, the range of an array is the _________.

Answer 21

number of linearly independent vectors in the array

Question 22

A system of two linear equations can have _________, an _____________, or _______. Systems of equations can be classified by ________.

Answer 22

one solution
infinite number of solutions
no solution

the number of solutions

Question 23

If a system has at least one solution, it is said to be __________.

Answer 23

consistent

Question 24

If a consistent system has exactly one solution, it is __________.

Answer 24

independent

Question 25

If a consistent system has an infinite number of solutions, it is ______. When you graph the equations, both equations represent the same line.

Answer 25

dependent

Question 26

If a system has no solution, it is said to be ____________. The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

Answer 26

inconsistent

Question 27

“Independent” means that each equation gives new information.
Otherwise they are “Dependent”.

Answer 27

True

Question 28

If the range of a 3x3 matrix is <3, the determinant is _____, and the matrix _________ an inverse, which means that it ___________ solution.

Answer 28

0
does not have
does not have a

Question 29

No solution means the graph of two linear equations are ____________.

Answer 29

parallel

Question 30

In order to find a general solution of a system of equations, one needs to simplify it as much as possible

Answer 30

True

Question 31

The simplest system of linear equations is one where every equation has only one unknown and all these unknowns are different.

Answer 31

True

Question 32

It is not possible to reduce every system of linear equations to this form, but we can get very close. There are three operations that one can apply to any system of linear equations: _______________

The system obtained after each of these operations is ____________ to the original system, meaning that they have the same solutions.

Answer 32

1. Replace an equation by the sum of this equation and another equation multiplied by a number
2. Swap two equations
3. Multiply an equation by a non-zero number

equivalent

Question 33

In order to simplify a system of equations, it is enough to simplify its _________ by using the following row operations:

Answer 33

augmented matrix

1. Replace a row by this row plus another row multiplied by a number.
2. Swap two rows.
3. Multiply a row by a non-zero number.

Question 34

We can get a general solution of every system of equations whose matrix is in the _________-.

Answer 34

reduced row echelon form

Question 35

A system of linear equations either has no solutions or has exactly one solution or has infinitely many solutions

Answer 35

True

Question 36

A system of linear equations has infinitely many solutions if and only if _______________.

Answer 36

its reduced row echelon form has free unknowns and the last column of the reduced row echelon form has no leading 1’s

Question 37

A system of linear equations has exactly one solution if and only if _______________.

Answer 37

the reduced row echelon form has no free unknowns and the last column of the reduced row echelon form has no leading 1

Question 38

A system of linear equations has no solutions if and only if the ______________.

Answer 38

last column of the reduced row echelon form has a leading 1

Question 39

In row echelon form, _____________, _______________, and _________________.

Answer 39

1. each leading entry must be 1
2. each leading entry must lie to the right of the leading entry above it
3. rows containing all zeros go last.

Question 40

Let V be a subspace of R^n. The number of vectors in any basis of V is called the dimension of V, and is written dim V

Answer 40

True

Question 41

A basis is in A is a set of vectors in A that:

Answer 41

1. Span A

2. Are linearly independent

Question 42

The dimension of A is the number of vectors in a ______________.

Answer 42

basis for A

Question 43

In subset ℝ^3, ℋ = {(x,y,z)∈ ℝ3 ǀ x+ y = 0, z + 2y = 0}, “ x+ y = 0, z + 2y = 0” is _______________.

Answer 43

the rule of the basis

Question 44

“Spans S” means that every vector in S is a scalar multiple of [x y … n].

Answer 44

True

Question 45

The linear combination of a single vector is a ______________.

Answer 45

scalar multiple

Question 46

The determinant of a triangular

matrix is the ________________ of its diagonal entries.

Answer 46

product

Question 47

A square matrix A is invertible if and only if _________.

Answer 47

detA ≠ 0

Question 48

det(AB) =

Answer 48

det(A) det(B)

Question 49

det(A) det(A−1) = det(AA−1) = det(I)=_____.

Answer 49

1

Question 50

If A is invertible, det(A^−1) = _________.

Answer 50

1/det A

Question 51

If Ax = λx for some scalar λ and some nonzero vector x,

then we say ______________ and x is _________.

Answer 51

λ is an eigenvalue of A

an eigenvector associated with λ

Question 52

Any (nonzero) scalar multiple of an eigenvector

is itself an eigenvector (associated w/same eigenvalue).

Answer 52

True

Question 53

Eigenvalues of A means find values ____________.

Answer 53

of λ such that det(A − λI) = 0

Question 54

Given eigenvalue λ, associated eigenvectors are

______________.

Answer 54

nonzero vectors in null(A − λI)