Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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 you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
