1 2 -1 2 -1 For what values of k will the matrix 3 be singular (singular = not invertible). -2 2 k
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.
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2.3 Invertible Matrices no lq.pdf
The theorem below combines many of the theorems that we have had so far in the course.
Theorem 2.3.1: The Invertible Matrix Theorem (IMT)
Let A be a square n xn matrix. Then the following statements are equivalent.
a) A is an invertible matrix.
b) A is row equivalent to I, (the n xn identity matrix).
c) A has n pivot positions (a pivot in every column and in every row).
d) The equation Ax = 0 has only the trivial solution.
e) The columns of A form a linearly independent set.
f) The linear transformation + AX is one-to-one.
g) The equation AX = b has exactly one solution for each b in R".
h) The columns of A span R".
i) The linear transformation - Ax maps R" onto R".
j) A" is an invertible matrix.
Video Explanation: https://youtu.be/hllaHn7zO14
Example 1:
3 -5
2 is invertible.
7)
Using as few calculations as possible, determine if A =
1
I-4 -9
Solution:
Since we are not asked to actually find, A-1, we do not need to set up the matrix [A | ).
Just reduce the matrix A:
3 -5 RRz
1
1
21
3 -5
15]
1
21
3 -5
2
3 -5
-4 -9
-4 -9
0 -9
At this point, we know that there is not a pivot in every row, so the matrix is not invertible by part c of the
Invertible Matrix Theorem.
Example 2:
A square matrix is said to be lower triangular if all entries above the main diagonal are zero.
[1 0 0
e.g. A = 2 3 o is lower triangular.
14 5 6
When is a lower triangular matrix invertible?
Solution:
[1 0 01
Consider A = 2 3
14. 5 61
[1 2 4
AT = 0 3 5. Since this matrix is in row echelon form, we know that AT has a pivot in every row.
Lo o 6
By part e of the Invertible Matrix Theorem, AT is invertible.
Then, by part į of the IMT, A is invertible.
However, if any entry on the main diagonal had been 0, the matrix would not be invertible.
[1 001
e.g. B = 2 0 0 is a lower triangular matrix.
14. 5 61
[1 2 4
B' = 0 0 5. This matrix does not have a pivot in every column, so it is not invertible.
lo o
o 6
Then B is not invertible.
Thus, a lower triangular matrix is invertible if and only if every entry on the main diagonal is nonzero.
Now try Written Assignment, Question 6:
1 2 -1
For what values of k will the matrix 2 -1 3 be singular (singular = not invertible).
-2 2
k](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4e23d50-b6e1-4880-9ad9-a789799b751b%2F20d076ba-77b4-47a4-a14c-69acf8b10613%2Fo8gfffp_processed.jpeg&w=3840&q=75)
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