Answer :
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].
