Is the equation Bx = p consistent for all p in R^4? Explain.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![**Understanding Matrix B and its Reduced Row Echelon Form**
Consider the matrix \( B \) and the reduced row echelon form (RREF) of \( B \).
The matrix \( B \) is:
\[
B = \begin{pmatrix}
1 & 4 & 2 \\
0 & 1 & -4 \\
0 & 2 & 7 \\
2 & 9 & -7
\end{pmatrix}
\]
To obtain the reduced row echelon form of \( B \), we perform row operations to simplify \( B \) into its echelon form and then proceed to further simplify to get the reduced row echelon form:
\[
B \xrightarrow[]{RREF} \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{pmatrix}
\]
### Explanation of the Reduced Row Echelon Form
The Reduced Row Echelon Form (RREF) of a matrix is a unique form where the matrix satisfies the following conditions:
1. Each leading entry in a row is 1.
2. Each leading 1 is the only nonzero entry in its column.
3. The leading 1 of a row is to the right of the leading 1 of the row above it.
4. Any row containing a leading 1 is above any row with all zero elements.
In this case, the RREF of \( B \) is an identity matrix with a row of zeros. This indicates that the matrix \( B \) has three pivotal rows and one non-pivotal row, indicating its rank is 3. This transformation simplifies solving systems of linear equations, among other applications.
By studying the transformation from matrix \( B \) to its RREF, you gain important insights into the nature and solution space of systems represented by the matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2F254cdd2b-730f-48fc-ba1d-247cfd38ce4c%2F4hlfpje_processed.jpeg&w=3840&q=75)
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