Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To solve this problem, let's approach each part step by step.
1. Find the inverse of [tex]\( M \)[/tex]:
Given matrix [tex]\( M \)[/tex]:
[tex]\[ M = \begin{pmatrix} 1 & 2 \\ -3 & -7 \end{pmatrix} \][/tex]
The inverse of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex] can be found using the formula:
[tex]\[ M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \][/tex]
For our matrix, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], [tex]\( c = -3 \)[/tex], [tex]\( d = -7 \)[/tex], so:
[tex]\[ ad - bc = (1 \cdot -7) - (2 \cdot -3) = -7 + 6 = -1 \][/tex]
Thus,
[tex]\[ M^{-1} = \frac{1}{-1} \begin{pmatrix} -7 & -2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 2 \\ -3 & -1 \end{pmatrix} \][/tex]
2. Show that [tex]\( M^{-1}M = I \)[/tex]:
To verify, we multiply [tex]\( M^{-1} \)[/tex] by [tex]\( M \)[/tex]:
[tex]\[ M^{-1}M = \begin{pmatrix} 7 & 2 \\ -3 & -1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ -3 & -7 \end{pmatrix} \][/tex]
Perform the matrix multiplication:
[tex]\[ \begin{pmatrix} 7 \cdot 1 + 2 \cdot -3 & 7 \cdot 2 + 2 \cdot -7 \\ -3 \cdot 1 + -1 \cdot -3 & -3 \cdot 2 + -1 \cdot -7 \end{pmatrix} = \begin{pmatrix} 7 - 6 & 14 - 14 \\ -3 + 3 & -6 + 7 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \][/tex]
This verifies that [tex]\( M^{-1} M \)[/tex] is indeed the identity matrix [tex]\( I \)[/tex].
3. Evaluate specific expressions:
- For [tex]\( (7 \cdot 1) + (2 \cdot -3) \)[/tex] corresponding to expression A:
[tex]\[ (7 \cdot 1) + (2 \cdot -3) = 7 - 6 = 1 \][/tex]
Therefore, the value in the first row, second column, corresponding to option A, is:
[tex]\[ \boxed{1} \][/tex]
- For [tex]\( (-3 \cdot 2) + (-1 \cdot -7) \)[/tex] corresponding to expression A:
[tex]\[ (-3 \cdot 2) + (-1 \cdot -7) = -6 + 7 = 1 \][/tex]
Therefore, the value in the second row, first column, corresponding to option A, is:
[tex]\[ \boxed{1} \][/tex]
In summary:
- The inverse of [tex]\( M \)[/tex] is [tex]\( \begin{pmatrix} 7 & 2 \\ -3 & -1 \end{pmatrix} \)[/tex].
- [tex]\( M^{-1} M \)[/tex] results in the identity matrix [tex]\( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)[/tex].
- The simplified value of expression A is [tex]\( 1 \)[/tex].
- The simplified value of expression A for the second row, first column, is [tex]\( 1 \)[/tex].
1. Find the inverse of [tex]\( M \)[/tex]:
Given matrix [tex]\( M \)[/tex]:
[tex]\[ M = \begin{pmatrix} 1 & 2 \\ -3 & -7 \end{pmatrix} \][/tex]
The inverse of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex] can be found using the formula:
[tex]\[ M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \][/tex]
For our matrix, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], [tex]\( c = -3 \)[/tex], [tex]\( d = -7 \)[/tex], so:
[tex]\[ ad - bc = (1 \cdot -7) - (2 \cdot -3) = -7 + 6 = -1 \][/tex]
Thus,
[tex]\[ M^{-1} = \frac{1}{-1} \begin{pmatrix} -7 & -2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 2 \\ -3 & -1 \end{pmatrix} \][/tex]
2. Show that [tex]\( M^{-1}M = I \)[/tex]:
To verify, we multiply [tex]\( M^{-1} \)[/tex] by [tex]\( M \)[/tex]:
[tex]\[ M^{-1}M = \begin{pmatrix} 7 & 2 \\ -3 & -1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ -3 & -7 \end{pmatrix} \][/tex]
Perform the matrix multiplication:
[tex]\[ \begin{pmatrix} 7 \cdot 1 + 2 \cdot -3 & 7 \cdot 2 + 2 \cdot -7 \\ -3 \cdot 1 + -1 \cdot -3 & -3 \cdot 2 + -1 \cdot -7 \end{pmatrix} = \begin{pmatrix} 7 - 6 & 14 - 14 \\ -3 + 3 & -6 + 7 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \][/tex]
This verifies that [tex]\( M^{-1} M \)[/tex] is indeed the identity matrix [tex]\( I \)[/tex].
3. Evaluate specific expressions:
- For [tex]\( (7 \cdot 1) + (2 \cdot -3) \)[/tex] corresponding to expression A:
[tex]\[ (7 \cdot 1) + (2 \cdot -3) = 7 - 6 = 1 \][/tex]
Therefore, the value in the first row, second column, corresponding to option A, is:
[tex]\[ \boxed{1} \][/tex]
- For [tex]\( (-3 \cdot 2) + (-1 \cdot -7) \)[/tex] corresponding to expression A:
[tex]\[ (-3 \cdot 2) + (-1 \cdot -7) = -6 + 7 = 1 \][/tex]
Therefore, the value in the second row, first column, corresponding to option A, is:
[tex]\[ \boxed{1} \][/tex]
In summary:
- The inverse of [tex]\( M \)[/tex] is [tex]\( \begin{pmatrix} 7 & 2 \\ -3 & -1 \end{pmatrix} \)[/tex].
- [tex]\( M^{-1} M \)[/tex] results in the identity matrix [tex]\( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)[/tex].
- The simplified value of expression A is [tex]\( 1 \)[/tex].
- The simplified value of expression A for the second row, first column, is [tex]\( 1 \)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
