Math / Algebra

QuestionWhich of the following are TRUE? [This question is based on your assigned pre-reading/prep for the upcoming Assignment] Transition matrices must be square. Entries in rows in a transition matrix add to 1. $\left[\begin{array}{ll}0.7 & 0.25 \\ 0.3 & 0.75\end{array}\right]$ could be a transition matrix. $\left[\begin{array}{cc}1.1 & -0.4 \\ -0.1 & 1.4\end{array}\right]$ could be a transition matrix.

Studdy Solution

STEP 1

What is this asking? Which statements about transition matrices are true?
Specifically, do they have to be square?
Do the rows add up to one?
And are the two given matrices valid transition matrices? Watch out! Don't mix up rows and columns!
A common mistake is to check if the columns sum to one instead of the rows.

STEP 2

1. Square Matrices
2. Row Sums
3. First Example Matrix
4. Second Example Matrix

STEP 3

Transition matrices must be square.
This is because they represent the probabilities of transitioning between states in a system.
If you have $n$ states, the matrix needs to be $n \times n$ to capture all possible transitions.

STEP 4

The entries in each row of a transition matrix must add up to $\textbf{1}$ .
Why? Because each row represents the probabilities of transitioning from a specific state to all other states (including itself!). Since something must happen, the probabilities along each row have to add up to one, representing 100% probability.

STEP 5

Let's look at the first matrix: $\begin{bmatrix} 0.7 & 0.25 \\ 0.3 & 0.75 \end{bmatrix}$

STEP 6

It's a square matrix, two by two, so that checks out!

STEP 7

Now, let's check the row sums.
The first row is $0.7 + 0.25 = 0.95$ .
The second row is $0.3 + 0.75 = 1.05$ .
Uh oh!
Neither of these rows adds up to $\textbf{1}$ .

STEP 8

So, even though it's square, this cannot be a transition matrix because the rows don't add up to $\textbf{1}$ .

STEP 9

Let's check out the second matrix: $\begin{matrix} 1.1 & -0.4 \\ -0.1 & 1.4 \end{matrix}$

STEP 10

It's square, two by two, so we're good so far.

STEP 11

Let's add up the rows!
The first row is $1.1 + (-0.4) = 0.7$ .
The second row is $(-0.1) + 1.4 = 1.3$ .
Neither of these add up to $\textbf{1}$ !

STEP 12

But wait, there's more!
Notice we have negative entries.
Probabilities cannot be negative!
They must be between $\textbf{0}$ and $\textbf{1}$ inclusive.
So, this definitely isn't a transition matrix.

STEP 13

Only the first statement, "Transition matrices must be square," is TRUE.
The rows must sum to one, not the columns.
Neither of the example matrices could be transition matrices.

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Studdy Solution

STEP 1

STEP 2

1. Square Matrices
2. Row Sums
3. First Example Matrix
4. Second Example Matrix

STEP 3

Transition matrices must be square.
This is because they represent the probabilities of transitioning between states in a system.
If you have $n$ states, the matrix needs to be $n \times n$ to capture all possible transitions.

STEP 4

STEP 5

Let's look at the first matrix: $\begin{bmatrix} 0.7 & 0.25 \\ 0.3 & 0.75 \end{bmatrix}$

STEP 6

It's a square matrix, two by two, so that checks out!

STEP 7

Now, let's check the row sums.
The first row is $0.7 + 0.25 = 0.95$ .
The second row is $0.3 + 0.75 = 1.05$ .
Uh oh!
Neither of these rows adds up to $\textbf{1}$ .

STEP 8

So, even though it's square, this cannot be a transition matrix because the rows don't add up to $\textbf{1}$ .

STEP 9

Let's check out the second matrix: $\begin{matrix} 1.1 & -0.4 \\ -0.1 & 1.4 \end{matrix}$

STEP 10

It's square, two by two, so we're good so far.

STEP 11

Let's add up the rows!
The first row is $1.1 + (-0.4) = 0.7$ .
The second row is $(-0.1) + 1.4 = 1.3$ .
Neither of these add up to $\textbf{1}$ !

STEP 12

But wait, there's more!
Notice we have negative entries.
Probabilities cannot be negative!
They must be between $\textbf{0}$ and $\textbf{1}$ inclusive.
So, this definitely isn't a transition matrix.

STEP 13

Only the first statement, "Transition matrices must be square," is TRUE.
The rows must sum to one, not the columns.
Neither of the example matrices could be transition matrices.

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Studdy Solution

STEP 1

STEP 2

1. Square Matrices2. Row Sums3. First Example Matrix4. Second Example Matrix

STEP 3

Transition matrices *must* be square.This is because they represent the probabilities of transitioning *between* states in a system.If you have nnn states, the matrix needs to be n×nn \times nn×n to capture all possible transitions.

STEP 4

STEP 5

Let's look at the first matrix: [0.70.250.30.75]\begin{bmatrix} 0.7 & 0.25 \\ 0.3 & 0.75 \end{bmatrix}[0.70.3​0.250.75​]

STEP 6

It's a square matrix, **two by two**, so that checks out!

STEP 7

Now, let's check the row sums.The first row is 0.7+0.25=0.950.7 + 0.25 = 0.950.7+0.25=0.95.The second row is 0.3+0.75=1.050.3 + 0.75 = 1.050.3+0.75=1.05.Uh oh!Neither of these rows adds up to 1\textbf{1}1.

STEP 8

So, even though it's square, this *cannot* be a transition matrix because the rows don't add up to 1\textbf{1}1.

STEP 9

Let's check out the second matrix: 1.1−0.4−0.11.4\begin{matrix} 1.1 & -0.4 \\ -0.1 & 1.4 \end{matrix}1.1−0.1​−0.41.4​

STEP 10

It's square, **two by two**, so we're good so far.

STEP 11

Let's add up the rows!The first row is 1.1+(−0.4)=0.71.1 + (-0.4) = 0.71.1+(−0.4)=0.7.The second row is (−0.1)+1.4=1.3(-0.1) + 1.4 = 1.3(−0.1)+1.4=1.3.Neither of these add up to 1\textbf{1}1!

STEP 12

But wait, there's more!Notice we have *negative* entries.Probabilities *cannot* be negative!They must be between 0\textbf{0}0 and 1\textbf{1}1 inclusive.So, this definitely isn't a transition matrix.

STEP 13

Only the first statement, "Transition matrices must be square," is **TRUE**.The rows must sum to one, not the columns.Neither of the example matrices could be transition matrices.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Square Matrices
2. Row Sums
3. First Example Matrix
4. Second Example Matrix

Transition matrices must be square.
This is because they represent the probabilities of transitioning between states in a system.
If you have $n$ states, the matrix needs to be $n \times n$ to capture all possible transitions.

Let's look at the first matrix: $\begin{bmatrix} 0.7 & 0.25 \\ 0.3 & 0.75 \end{bmatrix}$

It's a square matrix, two by two, so that checks out!

Now, let's check the row sums.
The first row is $0.7 + 0.25 = 0.95$ .
The second row is $0.3 + 0.75 = 1.05$ .
Uh oh!
Neither of these rows adds up to $\textbf{1}$ .

So, even though it's square, this cannot be a transition matrix because the rows don't add up to $\textbf{1}$ .

Let's check out the second matrix: $\begin{matrix} 1.1 & -0.4 \\ -0.1 & 1.4 \end{matrix}$

It's square, two by two, so we're good so far.

Let's add up the rows!
The first row is $1.1 + (-0.4) = 0.7$ .
The second row is $(-0.1) + 1.4 = 1.3$ .
Neither of these add up to $\textbf{1}$ !

But wait, there's more!
Notice we have negative entries.
Probabilities cannot be negative!
They must be between $\textbf{0}$ and $\textbf{1}$ inclusive.
So, this definitely isn't a transition matrix.

Only the first statement, "Transition matrices must be square," is TRUE.
The rows must sum to one, not the columns.
Neither of the example matrices could be transition matrices.