https://file.aiquickdraw.com/mathbot/file/17270964572546qetgytn.jpg

The image shows a matrix operation problem asking to determine the result of a scalar multiplication with a matrix. The matrix is:

\[
\begin{pmatrix}
2 & -1 & 8 \\
6 & 1 & 5
\end{pmatrix}
\]

On the right-hand side of the equation, the resulting matrix is shown as:

\[
\begin{pmatrix}
-10 & m \times 5 & -40 \\
-30 & -5 & -25
\end{pmatrix}
\]

We are tasked with determining the scalar \(m\) (which is missing) by comparing the original matrix to the result matrix, assuming the scalar multiplication has occurred.

### Solution:

Given that the matrix was multiplied by some scalar \(a\), we know the relationship between the entries of the original matrix and the resulting matrix. Let's analyze the corresponding elements to find the scalar:

1. For the element at position (1, 1):
   \[
   2a = -10 \quad \Rightarrow \quad a = \frac{-10}{2} = -5
   \]

2. Now that we know \(a = -5\), we can check the rest of the matrix to confirm:

- Element (1, 2): \(-5 \times -1 = 5\) (which confirms \(m = 1\) since the result shows \(5\))
- Element (1, 3): \(-5 \times 8 = -40\) (matches the result)
- Element (2, 1): \(-5 \times 6 = -30\) (matches the result)
- Element (2, 2): \(-5 \times 1 = -5\) (matches the result)
- Element (2, 3): \(-5 \times 5 = -25\) (matches the result)

Thus, the scalar \(a\) is \(-5\).

Let me know if you need more details on this solution or further questions!

### Related Questions:
1. How do we perform scalar multiplication of matrices in general?
2. What are the properties of scalar multiplication in matrix algebra?
3. How does matrix addition compare to scalar multiplication?
4. What is the determinant of the matrix after scalar multiplication?
5. How can scalar multiplication affect matrix transformations (e.g., scaling)?

**Tip:** When performing scalar multiplication, always distribute the scalar to **every element** in the matrix.

Determine the result of scalar multiplication with the following matrix: [[2, -1, 8], [6, 1, 5]] and [[-10, m*5, -40], [-30, -5, -25]]

Explore scalar multiplication of a matrix through an example. The matrix [[2, -1, 8], [6, 1, 5]] is multiplied by a scalar to yield [[-10, 5m, -40], [-30, -5, -25]]. Learn how to solve for the scalar and verify the result.

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