IDNLearn.com provides a seamless experience for finding accurate answers. Get comprehensive and trustworthy answers to all your questions from our knowledgeable community members.
Sagot :
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]
### 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]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.