https://file.aiquickdraw.com/mathbot/file/17296883356477x8i0iaq.jpg

The problem asks to find matrix \( G \) based on two given conditions. Let's analyze and solve the two parts one by one.

---

### Part (a):
The equation given is:

\[
\begin{pmatrix} 4 & 1 & 2 \\ -1 & 3 & 0 \end{pmatrix} + 2G = \begin{pmatrix} -2 & 1 & 4 \\ 9 & 1 & 6 \end{pmatrix}
\]

#### Steps:
1. **Isolate \( G \):**
   \[
   2G = \begin{pmatrix} -2 & 1 & 4 \\ 9 & 1 & 6 \end{pmatrix} - \begin{pmatrix} 4 & 1 & 2 \\ -1 & 3 & 0 \end{pmatrix}
   \]

   Calculate the matrix subtraction:
   \[
   \begin{pmatrix} -2 & 1 & 4 \\ 9 & 1 & 6 \end{pmatrix} - \begin{pmatrix} 4 & 1 & 2 \\ -1 & 3 & 0 \end{pmatrix} = \begin{pmatrix} -6 & 0 & 2 \\ 10 & -2 & 6 \end{pmatrix}
   \]

2. **Divide by 2 to find \( G \):**
   \[
   G = \frac{1}{2} \begin{pmatrix} -6 & 0 & 2 \\ 10 & -2 & 6 \end{pmatrix} = \begin{pmatrix} -3 & 0 & 1 \\ 5 & -1 & 3 \end{pmatrix}
   \]

Thus, the solution for part (a) is:
\[
G = \begin{pmatrix} -3 & 0 & 1 \\ 5 & -1 & 3 \end{pmatrix}
\]

---

### Part (b):
The equation given is:

\[
3G - 2\begin{pmatrix} 1 & 1 & 0 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix}
\]

#### Steps:
1. **Expand the equation:**
   \[
   3G - 2\begin{pmatrix} 1 & 1 & 0 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix}
   \]

   Calculate the multiplication:
   \[
   2\begin{pmatrix} 1 & 1 & 0 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 2 & 0 \\ -4 & 4 & 0 \end{pmatrix}
   \]

   Thus, the equation becomes:
   \[
   3G - \begin{pmatrix} 2 & 2 & 0 \\ -4 & 4 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix}
   \]

2. **Isolate \( 3G \):**
   \[
   3G = \begin{pmatrix} 1 & 6 & 5 \\ -2 & -1 & 0 \end{pmatrix} + \begin{pmatrix} 2 & 2 & 0 \\ -4 & 4 & 0 \end{pmatrix}
   \]

   Add the matrices:
   \[
   3G = \begin{pmatrix} 3 & 8 & 5 \\ -6 & 3 & 0 \end{pmatrix}
   \]

3. **Divide by 3 to find \( G \):**
   \[
   G = \frac{1}{3} \begin{pmatrix} 3 & 8 & 5 \\ -6 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 1 & \frac{8}{3} & \frac{5}{3} \\ -2 & 1 & 0 \end{pmatrix}
   \]

Thus, the solution for part (b) is:
\[
G = \begin{pmatrix} 1 & \frac{8}{3} & \frac{5}{3} \\ -2 & 1 & 0 \end{pmatrix}
\]

---

Would you like further details on any steps or have questions about matrix operations?

Here are 5 follow-up questions you could explore:
1. What are some properties of matrix addition and subtraction?
2. How does matrix scalar multiplication work?
3. Can we find the determinant of matrix \( G \) in part (a)?
4. How do we verify the solutions by substituting \( G \) back into the original equations?
5. What is the significance of the identity matrix in matrix equations?

**Tip:** When working with matrices, always ensure that the dimensions match for addition, subtraction, or multiplication operations.

Tentukan matriks G jika: (a) (4 1 2) (-1 3 0) + 2G = (-2 1 4) (9 1 6), (b) 3G - 2(1 1 0) (-2 2 0) = (1 6 5) (-2 -1 0)

This problem involves solving for matrix G based on matrix equations with matrix addition, scalar multiplication, and subtraction. The problem tests knowledge of matrix operations suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation