-1 Find all values of x such that the matrix 8 is not invertible.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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
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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