Simultaneous Equations

Question 1

If a system of equations has no solutions, what is it called?

Answer 1

Inconsistent

Question 2

When at least one solution exists for a given system of linear equations, we call that system ________.

Answer 2

When at least one solution exists for a given system of linear equations, we call that system consistent .

Question 3

A consistent system of linear equations has XXXXXX solutions.

Answer 3

A consistent system of linear equations has One or An Infinite solutions.

Question 4

If a consistent system has an infinite number of solutions, if we can define a solution in terms of some extra parameter t , we call this a XXXXXXXXX solution .

Answer 4

If a consistent system has an infinite number of solutions, if we can define a solution in terms of some extra parameter t , we call this a parametric solution .

Question 5

What are three definitions for the Rank of a matrix?

Answer 5

The rank of a matrix is any of:

1. the number of pivots when the matrix is in ref
2. the number of independent rows
3. the number of independent columns

All three are equivalent.

Question 6

What is the definition of the Row Canonical Form of a matrix?

Answer 6

A matrix is in row canonical form if when:

• the pivot in each row is a 1
• the pivot is the only non-zero entry in its column.

Question 7

How do you solve a system of simultaneous equations using matrices?

Answer 7

• Write the equations in matrix form Ax=b
• Create an augmented matrix from A and b
• Reduce the matrix to ref or rref form
• Use back substiution to solve for each variable

(if the matrix is not full rank you will have to create a parametric solution using the free variables)

Question 8

What are Elementary Matrices?

Answer 8

They are square matrices that represent the elementary row operations.

Question 9

What does an Elementary Matrix that represents swapping two rows look like?

Answer 9

Start with an apppropriately sized Identity matrix and then swap the same two rows that you want swapped in the target matrix.

Question 10

What does an Elementary Matrix that represents scaling a row by a constant look like?

Answer 10

Start with an apppropriately sized Identity matrix and then replace the 1 in the row you want scaled with the value you want it scaled by.

Question 11

What does an elementary matrix look like that adds some multiple of one row to another row?

Answer 11

1. Start with an identity matrix I of the appropriate size.
2. Change the rows of ‘I’ to relflect the changes you want in the target matrix. Call this matrix D.
3. Mulitply the target matrix by the left by D (DA)

Question 12

What does it mean if a matrix is singular?

Answer 12

A Singular matrix in non-invertible

Question 13

Is an invertible matrix singular or non-singular?

Answer 13

Invertible = non-singular

Question 14

When solving Ax=b using an augmented matrix {Ab] (where A is a square matrix),

what does it mean when the last row of the ref form takes the form of

[0 0 0…0 | n] where n is non-zero?

Answer 14

It means the equation Ax=b has no solutions.

Question 15

When solving Ax=b using an augmented matrix {Ab],

what does it mean when the last row of the ref form takes the form of

[0 0 0…0 | 0] ?

Answer 15

The Ax=b will have an infinite number of solutions

Question 16

For a Homogeneous Set of Equations,

if in the rref form

the # of equations = # of unknowns

How many solutions are there to the system?

Answer 16

Only one solution:

The zero solution [0 0 0 … 0]

Question 17

What is the general form of solutions for a system of simultaneous equations?

Answer 17

in general, a solution has the form of

1. a vector that is a particular solution of the system
2. added to an unrestricted combination of some other vectors.

Question 18

What makes an equation homogeneous?

Answer 18

A linear equation is homogeneous if it has a constant of zero, that is, if it can be put in the form show below.

Question 19

Why is it advantageous to study a homogeneous system of eqations that are associated with a non-homogeneous system of equations?

Answer 19

Homogeneous equations always have at least one solution - the zero vector.

Studying the associated homogeneous system has a great advantage over studying the original system. Nonhomogeneous systems can be inconsistent. But a homogeneous system must be consistent since there is always at least one solution, the vector of zeros.

Question 20

What is the formal theorem of the

General Solution = Particular + Homogeneous solutions

Answer 20

Question 21

What are the names of the two parts of the solution of a system of simultaneous equations?

Answer 21

The General Solution = the Particular solution + the Homogeneous solution

Question 22

If a set of homogeneous equations has only one solution, what will that solution be?

Answer 22

The 0 vector.

Question 23

How many solutions can a set of homogeneous equations have?

Answer 23

Either

• one solution (the 0 vector) or
• many solutions (including the 0 vector)

Question 24

How many solution vectors will there be to a homogeneous set of equations?

Answer 24

There will be as many solution vectors as there is free variables in the RREF of the equations.

Question 25

When thinking about the General solution to a set of simultaneous equations, when the Homogeous part of the solution has only one solution, what can you way about the Homogeneous part of the solution?

Answer 25

The homogeneous part of the solution is the 'em[ty set' of vectors. In other words, the homogeneous part doesn't add anything to the Particular solution.

Question 26

Why do we go to echelon form as a particularly simple, or basic, version of a linear system?

Answer 26

The answer, of course, is that echelon form is suitable for back substitution, because we have isolated the variables.

Question 27

What is the row space of a matrix?

Answer 27

The row space of a matrix is the set of all possible linear combinations of its row vectors.

Question 28

Do Elementary Row operations effect the row space of a matrix?

Answer 28

No. ## Footnote A matrix that has been row reduced to either REF or RREF form will have the same row space as the original matrix.

Question 29

What condition must exist for two matrices to be row equivalent?

Answer 29

Two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations.

Question 30

If two matrices of the same size are row equivalent, what does that mean about the solutions to the two matrices corresponding homogenous systems?

Answer 30

Two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions.

Question 31

A matrix is invertible if and only if it is row equivalent to -------------------------.

Answer 31

A matrix is invertible if and only if it is row equivalent to **the identity matrix**.

Question 32

In an echelon form matrix, no nonzero row is a ------ -------- of the other rows.

Answer 32

In an echelon form matrix, no nonzero row is a **linear combination** of the other rows.