To solve this problem, we'll go through several steps. We'll start by calculating the matrix [tex]\(2M + 3N\)[/tex], then find its transpose, and finally determine the determinant of the transposed matrix.
### Step 1: Calculate [tex]\(2M\)[/tex]
Given matrix [tex]\(M\)[/tex]:
[tex]\[ M = \begin{pmatrix} 3 & 4 \\ -2 & 5 \end{pmatrix} \][/tex]
First, calculate [tex]\(2M\)[/tex]:
[tex]\[
2M = 2 \times \begin{pmatrix} 3 & 4 \\ -2 & 5 \end{pmatrix} = \begin{pmatrix} 2 \times 3 & 2 \times 4 \\ 2 \times -2 & 2 \times 5 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ -4 & 10 \end{pmatrix}
\][/tex]
### Step 2: Calculate [tex]\(3N\)[/tex]
Given matrix [tex]\(N\)[/tex]:
[tex]\[ N = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix} \][/tex]
Next, calculate [tex]\(3N\)[/tex]:
[tex]\[
3N = 3 \times \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 3 \times 2 & 3 \times 1 \\ 3 \times -1 & 3 \times 3 \end{pmatrix} = \begin{pmatrix} 6 & 3 \\ -3 & 9 \end{pmatrix}
\][/tex]
### Step 3: Calculate [tex]\(2M + 3N\)[/tex]
Now, sum [tex]\(2M\)[/tex] and [tex]\(3N\)[/tex]:
[tex]\[
2M + 3N = \begin{pmatrix} 6 & 8 \\ -4 & 10 \end{pmatrix} + \begin{pmatrix} 6 & 3 \\ -3 & 9 \end{pmatrix} = \begin{pmatrix} 6 + 6 & 8 + 3 \\ -4 + (-3) & 10 + 9 \end{pmatrix} = \begin{pmatrix} 12 & 11 \\ -7 & 19 \end{pmatrix}
\][/tex]
### Step 4: Transpose the matrix [tex]\((2M + 3N)\)[/tex]
To find the transpose of the matrix, interchange its rows and columns:
Let
[tex]\[ A = 2M + 3N = \begin{pmatrix} 12 & 11 \\ -7 & 19 \end{pmatrix} \][/tex]
Then,
[tex]\[ A^T = \begin{pmatrix} 12 & -7 \\ 11 & 19 \end{pmatrix} \][/tex]
### Step 5: Calculate the determinant of [tex]\(A^T\)[/tex]
The determinant of a 2x2 matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is calculated as [tex]\( ad - bc \)[/tex].
For [tex]\(A^T = \begin{pmatrix} 12 & -7 \\ 11 & 19 \end{pmatrix} \)[/tex]:
[tex]\[
\text{det}(A^T) = (12 \times 19) - (-7 \times 11) = 228 + 77 = 305
\][/tex]
### Summary
The matrix [tex]\((2M + 3N)^T\)[/tex] is:
[tex]\[ \begin{pmatrix} 12 & -7 \\ 11 & 19 \end{pmatrix} \][/tex]
The determinant of this transposed matrix is:
[tex]\[ 305 \][/tex]
Therefore, the final answer includes the transposed matrix and its determinant:
[tex]\[
\left( (2M + 3N)^T = \begin{pmatrix} 12 & -7 \\ 11 & 19 \end{pmatrix} \right)
\][/tex]
[tex]\[
\text{Determinant} = 305
\][/tex]
