Calculate the magnitude of vector (3,4), find the resultant of (2,3) and (4,1), add matrices [[1,2],[3,4]] and [[2,0],[1,5]], and find the determinant of [[1,2],[3,4]].

$$
\text{Magnitude} = \sqrt{3^2 + 4^2}
$$
$$
\text{Magnitude} = \sqrt{9 + 16}
$$
$$
\text{Magnitude} = \sqrt{25}
$$
$$
\text{Magnitude} = 5
$$
The magnitude of the vector is $\boxed{5}$.

Let $\mathbf{v_1} = (2,3)$ and $\mathbf{v_2} = (4,1)$.

$$
\text{Resultant} = (2+4, 3+1)
$$
$$
\text{Resultant} = (6,4)
$$
The resultant vector is $\boxed{(6,4)}$.

$$
\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix} = \begin{bmatrix} 1+2 & 2+0 \\ 3+1 & 4+5 \end{bmatrix}
$$

$$
= \begin{bmatrix} 3 & 2 \\ 4 & 9 \end{bmatrix}
$$
The sum of the matrices is $\boxed{\begin{bmatrix} 3 & 2 \\ 4 & 9 \end{bmatrix}}$.

$$
\text{Determinant} = (2 \times 4) - (3 \times 1)
$$
$$
\text{Determinant} = 8 - 3
$$
$$
\text{Determinant} = 5
$$
The determinant of the matrix is $\boxed{5}$.

Total balls = Number of red balls + Number of blue balls  
Total balls = $3 + 2 = 5$ balls.

Number of red balls = 3.

$$
\text{P(Picking Red)} = \frac{\text{Number of red balls}}{\text{Total number of balls}}
$$
$$
\text{P(Picking Red)} = \frac{3}{5}
$$
The probability of picking a red ball is $\boxed{\frac{3}{5}}$.