Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Concept explainers

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 …
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 choosi…
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 prob…
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Written Assignment Question 3? Please solve in details.
3:23 0 & A all 11%I 2.2 Inverse of a Matrix.pdf Theorem 2.2.1: Let A = If ad - be +0, then A is invertible and d -b ad - be If ad - be = 0, then A is not invertible. Note that in Example 3, for A = 0 . ad – bc = 0(1) – 1(0) = 0, so the matrix is not invertible. (We proved this another way.) > Note: We have a special name for the expression ad – bc. This is called the determinant of A. The following definition makes this more precise. DEFINITION: Determinant of a 2x2 Matrix Let A = 1 The determinant of A is defined as follows: det(A) = ad - be. Example 4: Find the inverse of the matrix = . if A is invertible. Solution: First, we calculate det (A) to make sure matrix A is invertible: det(A) = (3)(6) - (4)(5) = -2, so A is invertible. Next, we apply the formula from Theorem 2.2.1, which gives: A1 =L - -3/2) Link to Video Solution by Professor Matt Lewis: httns://www.educreations.com/lesson/view/example-2-2- 3-1/32439987/ Example 5: Find the inverse of the matrix = : if A is invertible. Solution: First, we calculate det (A) to make sure matrix A is invertible: det(A) = (-2)(8) – (4)(-4) = -16 - (-16) = -16 + 16 = 0, so A is not invertible. Now try Written Assignment, Question 3: Find all values of x such that the matrix [,. -1 .0 , 48 x- 3 |is not invertible. IV. Using A-1 to Solve the Matrix Equation Ax = b Theorem 2.2.2: If A is an invertible n xn matrix, then for each b in R", the equation AX = b has the unique solution = A-b. Idea: A = 6 A(AR) = A-b (A'A) = A-b I = A-5 Example 6: Consider the linear systems below. (a) (b) (c) [3x1 + 4x2 = 3 15х, + 6х, 3D 7 (3x, + 4x, = 10 15х, + 6х, — —2 (3x1 + 4x2 = 1 (5х, + 6x, 3
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Transcribed Image Text:3:23 0 & A all 11%I 2.2 Inverse of a Matrix.pdf Theorem 2.2.1: Let A = If ad - be +0, then A is invertible and d -b ad - be If ad - be = 0, then A is not invertible. Note that in Example 3, for A = 0 . ad – bc = 0(1) – 1(0) = 0, so the matrix is not invertible. (We proved this another way.) > Note: We have a special name for the expression ad – bc. This is called the determinant of A. The following definition makes this more precise. DEFINITION: Determinant of a 2x2 Matrix Let A = 1 The determinant of A is defined as follows: det(A) = ad - be. Example 4: Find the inverse of the matrix = . if A is invertible. Solution: First, we calculate det (A) to make sure matrix A is invertible: det(A) = (3)(6) - (4)(5) = -2, so A is invertible. Next, we apply the formula from Theorem 2.2.1, which gives: A1 =L - -3/2) Link to Video Solution by Professor Matt Lewis: httns://www.educreations.com/lesson/view/example-2-2- 3-1/32439987/ Example 5: Find the inverse of the matrix = : if A is invertible. Solution: First, we calculate det (A) to make sure matrix A is invertible: det(A) = (-2)(8) – (4)(-4) = -16 - (-16) = -16 + 16 = 0, so A is not invertible. Now try Written Assignment, Question 3: Find all values of x such that the matrix [,. -1 .0 , 48 x- 3 |is not invertible. IV. Using A-1 to Solve the Matrix Equation Ax = b Theorem 2.2.2: If A is an invertible n xn matrix, then for each b in R", the equation AX = b has the unique solution = A-b. Idea: A = 6 A(AR) = A-b (A'A) = A-b I = A-5 Example 6: Consider the linear systems below. (a) (b) (c) [3x1 + 4x2 = 3 15х, + 6х, 3D 7 (3x, + 4x, = 10 15х, + 6х, — —2 (3x1 + 4x2 = 1 (5х, + 6x, 3
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