If \begin{bmatrix} 11&2\\ 7&13 \end{bmatrix}\cdot G= \begin{bmatrix} \dfrac{-5}{4}&14\\ \dfrac{-97}{4}&91 \end{bmatrix}, then G=\begin{bmatrix} g_{1}&g_{2}\\ g_{3}&g_{4} \end{bmatrix}.

What is the value of g_{2}?

(It's okay to use a calculator to find the necessary inverse and multiply.)

We are given the matrix equation:

\[
\begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix} \cdot G = \begin{bmatrix} \dfrac{-5}{4} & 14 \\ \dfrac{-97}{4} & 91 \end{bmatrix},
\]

where \( G = \begin{bmatrix} g_1 & g_2 \\ g_3 & g_4 \end{bmatrix} \).

To solve for \( G \), we need to isolate it by multiplying both sides of the equation by the inverse of the matrix \( A = \begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix} \).

### Step 1: Find the inverse of \( A \)

The formula for the inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is:

\[
A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.
\]

For \( A = \begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix} \):

- \( a = 11 \), \( b = 2 \), \( c = 7 \), \( d = 13 \),
- Determinant \( \det(A) = ad - bc = 11 \times 13 - 7 \times 2 = 143 - 14 = 129 \).

Thus, the inverse is:

\[
A^{-1} = \frac{1}{129} \begin{bmatrix} 13 & -2 \\ -7 & 11 \end{bmatrix} = \begin{bmatrix} \frac{13}{129} & \frac{-2}{129} \\ \frac{-7}{129} & \frac{11}{129} \end{bmatrix}.
\]

### Step 2: Multiply \( A^{-1} \) by the matrix on the right-hand side

Now, to find \( G \), we compute:

\[
G = A^{-1} \cdot \begin{bmatrix} \dfrac{-5}{4} & 14 \\ \dfrac{-97}{4} & 91 \end{bmatrix}.
\]

We will focus on finding \( g_2 \), which is the element in the first row, second column of the resulting matrix.

\[
g_2 = \left( \frac{13}{129} \right) \times 14 + \left( \frac{-2}{129} \right) \times 91.
\]

### Step 3: Calculate \( g_2 \)

First, calculate each term:

- \( \frac{13}{129} \times 14 = \frac{13 \times 14}{129} = \frac{182}{129} \),
- \( \frac{-2}{129} \times 91 = \frac{-2 \times 91}{129} = \frac{-182}{129} \).

Now, sum the two terms:

\[
g_2 = \frac{182}{129} + \frac{-182}{129} = 0.
\]

Thus, the value of \( g_2 \) is \( \boxed{0} \).

---

Do you want further details or have any questions about this? 

Here are five related questions:
1. How is the determinant of a 2x2 matrix calculated?
2. Can all 2x2 matrices be inverted?
3. What happens if the determinant of a matrix is zero?
4. How do you multiply two matrices together?
5. What are some properties of matrix inverses?

**Tip**: Always check that the determinant is non-zero before attempting to find the inverse of a matrix!

If \( \begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix} \cdot G = \begin{bmatrix} \dfrac{-5}{4} & 14 \\ \dfrac{-97}{4} & 91 \end{bmatrix} \), then \( G = \begin{bmatrix} g_{1} & g_{2} \\ g_{3} & g_{4} \end{bmatrix} \). What is the value of \( g_{2} \)?

Solve the matrix equation \( \begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix} \cdot G = \begin{bmatrix} \dfrac{-5}{4} & 14 \\ \dfrac{-97}{4} & 91 \end{bmatrix} \). Learn how to compute the inverse of a 2x2 matrix, multiply matrices, and find the value of \( g_2 \) using step-by-step calculations. Suitable for college students studying linear algebra.

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