Let's work through the problem step by step.
1. Given Matrix [tex]\( M \)[/tex]:
[tex]\[
M = \begin{pmatrix}
1 & 2 \\
-3 & -7
\end{pmatrix}
\][/tex]
2. Find the Inverse of Matrix [tex]\( M \)[/tex], denoted as [tex]\( M^{-1} \)[/tex]:
The inverse of a [tex]\(2 \times 2\)[/tex] matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is calculated using the formula:
[tex]\[
\frac{1}{ad - bc} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\][/tex]
Applying this formula to our matrix [tex]\( M \)[/tex]:
[tex]\[
a = 1, \, b = 2, \, c = -3, \, d = -7
\][/tex]
Then,
[tex]\[
ad - bc = (1 \cdot -7) - (2 \cdot -3) = -7 + 6 = -1
\][/tex]
The inverse matrix [tex]\( M^{-1} \)[/tex] is:
[tex]\[
M^{-1} = \frac{1}{-1} \begin{pmatrix}
-7 & -2 \\
3 & 1
\end{pmatrix} = \begin{pmatrix}
7 & 2 \\
-3 & -1
\end{pmatrix}
\][/tex]
3. Find the product [tex]\( M^{-1}M \)[/tex]:
Next, we take the product of [tex]\( M^{-1} \)[/tex] and [tex]\( M \)[/tex]:
[tex]\[
M^{-1}M = \begin{pmatrix}
7 & 2 \\
-3 & -1
\end{pmatrix}
\begin{pmatrix}
1 & 2 \\
-3 & -7
\end{pmatrix}
\][/tex]
4. Matrix Multiplication:
To find the element in the first row and first column, we compute:
[tex]\[
(7 \cdot 1) + (2 \cdot -3) = 7 + (-6) = 1 \\
\][/tex]
To find the element in the first row and second column, we compute:
[tex]\[
(7 \cdot 2) + (2 \cdot -7) = 14 + (-14) = 0
\][/tex]
5. Summary of Results:
- The element in the first row and first column of [tex]\( M^{-1}M \)[/tex]:
[tex]\[
1
\][/tex]
The corresponding correct expression is:
B. [tex]\( (7 \cdot 1) + (2 \cdot -3) = 1 \)[/tex]
- The element in the first row and second column of[tex]\( M^{-1}M \)[/tex]:
[tex]\[
0
\][/tex]
The corresponding correct expression is:
D. [tex]\( (7 \cdot 2) + (2 \cdot -7) = 0 \)[/tex]
