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The given matrix \( B \) is:

\[
B = \begin{pmatrix} u & 3 & 1 \\ -2 & v & 4 \end{pmatrix}
\]

You are asked to determine the values of \( u \) and \( v \) based on two conditions:

### Condition a:
\[
3b_{11} = 6b_{23} \quad \text{and} \quad 2b_{22} = 4b_{21}
\]
Translating into elements of matrix \( B \):
- \( b_{11} = u \), \( b_{23} = 4 \), so the first equation becomes:
  \[
  3u = 6 \times 4 \quad \Rightarrow \quad 3u = 24 \quad \Rightarrow \quad u = 8
  \]
- \( b_{22} = v \), \( b_{21} = -2 \), so the second equation becomes:
  \[
  2v = 4 \times (-2) \quad \Rightarrow \quad 2v = -8 \quad \Rightarrow \quad v = -4
  \]

Thus, for condition (a), \( u = 8 \) and \( v = -4 \).

### Condition b:
\[
2b_{11} - 4b_{22} = 6 \quad \text{and} \quad b_{22} = b_{13}
\]
Using the matrix elements:
- \( b_{11} = u \), \( b_{22} = v \), so the first equation becomes:
  \[
  2u - 4v = 6
  \]
  Substituting \( u = 8 \) and \( v = -4 \):
  \[
  2(8) - 4(-4) = 6 \quad \Rightarrow \quad 16 + 16 = 6 \quad \Rightarrow \quad 32 \neq 6
  \]
  This equation is not valid with \( u = 8 \) and \( v = -4 \), meaning condition (b) introduces a contradiction, and further clarification might be needed for condition (b) to hold.

Would you like further clarification or to explore other conditions?

Here are 5 related questions for further exploration:
1. What if \( u \) or \( v \) had different values for condition (b)?
2. How would changing the matrix entries affect the system of equations?
3. Can matrix inversion be used to check consistency of the system?
4. What role do determinants play in solving matrix equations like these?
5. Could we use linear algebra tools like Gaussian elimination to solve this?

**Tip:** When solving systems of equations, double-check for contradictions across different conditions or constraints.

Diketahui matriks B = [[u, 3, 1], [-2, v, 4]]. Tentukan nilai u dan v jika: a) 3b11 = 6b23 dan 2b22 = 4b21, b) 2b11 - 4b22 = 6 dan b22 = b13.

Learn how to solve for values of u and v in a 2x3 matrix given algebraic conditions. This problem focuses on solving systems of equations derived from matrix entries, suitable for students in Grades 10-12.

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