We are given a matrix equation: $[4 \ 3] \begin{bmatrix} 1 & -2m \\ 3 & -1 \end{bmatrix} = [13 \ -35]$. We need to find the value of $m$.

1. Problem Description

We are given a matrix equation:

$[4 \ 3] \begin{bmatrix} 1 & -2m \\ 3 & -1 \end{bmatrix} = [13 \ -35]$.

We need to find the value of $m$.

2. Solution Steps

First, perform the matrix multiplication on the left-hand side:

$[4 \ 3] \begin{bmatrix} 1 & -2m \\ 3 & -1 \end{bmatrix} = [4(1) + 3(3) \ \ 4(-2m) + 3(-1)]$

$[4 \ 3] \begin{bmatrix} 1 & -2m \\ 3 & -1 \end{bmatrix} = [4 + 9 \ \ -8m - 3]$

$[4 \ 3] \begin{bmatrix} 1 & -2m \\ 3 & -1 \end{bmatrix} = [13 \ \ -8m - 3]$

Now we have the equation:

$[13 \ \ -8m - 3] = [13 \ -35]$

Equating the corresponding entries gives us two equations:

$13 = 13$

$-8m - 3 = -35$

We only need the second equation to find the value of $m$.

$-8m - 3 = -35$

$-8m = -35 + 3$

$-8m = -32$

$m = \frac{-32}{-8}$

$m = 4$

3. Final Answer

$m = 4$