Now try Written Assignment, Question 4: 1 Find the inverse of the matrix A =2 -3 2 3 -2 5 -3 by performing row-reduction on the matrix [A | ). -41
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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A ll 11%I
2.2 Inverse of a Matrix.pdf
AB = : -6 9
(-a + 4c -b + 4d
12a + 3c 2b + 3al = l6 9
Looking at column 1, we have:
-a + 4c = 1
2a + 3c = 0
Looking at column 2, we have:
-b + 4d = 0
2b + 3d = 1
Which can be solved by reducing the matrices:
and
These can be combined into a single matrix (similar to the example above):
Then bring this matrix to RREF. If you obtain a matrix of the form
6 1:), then the right side of this matrix is the solution. That is, the right side is A-1.
Summary:
Let A be an n xn matrix. To find A-1:
Adjoin A to I, to obtain the matrix [A | J.
Bring the matrix to row reduced echelon form.
a. If the RREF looks like [ I | B), then A-1 = B.
b. If the left side of the reduced echelon form is not Ig, then A is not invertible.
1.
2.
Example 7:
[o 1 21
3
Find the inverse of the matrix A =
14
-3 81
Solution:
Set up the augmented matrix for [A 1], which is
so 1 2 1 0 o1
1
0 30 1 0
[4 ]
-3 8 0 0 1
Row reduce the above matrix until lz appears on the left hand side, which yields
[1 0 0 -9/2
7
-3/21
0 10
-2
-1
lo 0 1
3/2
-2
1/2
Link to Video Solution by Professor Matt Lewis:
https://www.educreations.com/lesson/view/example-2-2-4-1/32440060/
Now try Written Assignment, Question 4:
3 -21
Find the inverse of the matrix A = 2 5 -3 by performing row-reduction on the matrix [A | 13].
1
L-3 2 -4.
Theorem 2.2.3:
If A and B are invertible n xn matrices, then:
• (A-1)-1 = A
• (AB)-1 = B-A-1
Proof that (AB)-' = B-'A-:
Let C = (AB)-1
Since (AB)(AB)-1 = 1, then (AB)C = I.
Multiply both sides of the equation on the left by A-1:
Regrouping the terms on the left side using the associative property:
A-(AB)C = A-.
(A-'A)BC = A-
Then:
IBC = A-1.
BC = A-1
B-'BC = B-A-1
Multiplying both sides of the equation on the left by B-1:
IC = B-'A-1
so C = B-1A-1
Thus, C = (AB)-1 = B-'A-
Example 8:
Solve the equation below for X, assuming A, B, and C are n x n invertible matrices:
C(B + X)A-1 = In](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4e23d50-b6e1-4880-9ad9-a789799b751b%2F4c7c7b5b-1e7b-4100-95c9-03be81b0f97d%2Furk5dp8_processed.jpeg&w=3840&q=75)
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