Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
All right, let's solve the problem step by step.
We are given two matrices [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] and we need to find the determinant of the matrix [tex]\(\frac{1}{2}P + Q\)[/tex].
Matrix [tex]\( P \)[/tex] is:
[tex]\[ P = \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} \][/tex]
Matrix [tex]\( Q \)[/tex] is:
[tex]\[ Q = \begin{pmatrix} 2 & 4 \\ 6 & 2 \end{pmatrix} \][/tex]
First, let's compute the determinant of [tex]\( P \)[/tex].
### Step 1: Determinant of [tex]\( P \)[/tex]
For a 2x2 matrix [tex]\( P = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the determinant is given by:
[tex]\[ \text{det}(P) = ad - bc \][/tex]
Substituting the values from matrix [tex]\( P \)[/tex]:
[tex]\[ \text{det}(P) = (2 \cdot 1) - (0 \cdot 1) = 2 - 0 = 2 \][/tex]
So, the determinant of [tex]\( P \)[/tex] is:
[tex]\[ \text{det}(P) = 2 \][/tex]
### Step 2: Determinant of [tex]\( Q \)[/tex]
Similarly, we compute the determinant of [tex]\( Q \)[/tex].
For matrix [tex]\( Q = \begin{pmatrix} 2 & 4 \\ 6 & 2 \end{pmatrix} \)[/tex]:
[tex]\[ \text{det}(Q) = (2 \cdot 2) - (4 \cdot 6) = 4 - 24 = -20 \][/tex]
So, the determinant of [tex]\( Q \)[/tex] is:
[tex]\[ \text{det}(Q) = -20 \][/tex]
### Step 3: Calculate Matrix [tex]\(\frac{1}{2}P + Q\)[/tex]
First, we need to compute [tex]\(\frac{1}{2}P\)[/tex].
[tex]\[ \frac{1}{2}P = \frac{1}{2} \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \][/tex]
Now, add this result to matrix [tex]\( Q \)[/tex].
[tex]\[ \frac{1}{2}P + Q = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} + \begin{pmatrix} 2 & 4 \\ 6 & 2 \end{pmatrix} = \begin{pmatrix} 1 + 2 & 0 + 4 \\ \frac{1}{2} + 6 & \frac{1}{2} + 2 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 6.5 & 2.5 \end{pmatrix} \][/tex]
### Step 4: Determinant of [tex]\(\frac{1}{2}P + Q\)[/tex]
Now we compute the determinant of the new matrix [tex]\( \frac{1}{2}P + Q \)[/tex].
For matrix [tex]\( \begin{pmatrix} 3 & 4 \\ 6.5 & 2.5 \end{pmatrix} \)[/tex]:
[tex]\[ \text{det}\left(\frac{1}{2}P + Q\right) = (3 \cdot 2.5) - (4 \cdot 6.5) = 7.5 - 26 = -18.5 \][/tex]
### Final Results:
- The determinant of [tex]\( P \)[/tex] is:
[tex]\[ \text{det}(P) = 2 \][/tex]
- The determinant of [tex]\( Q \)[/tex] is:
[tex]\[ \text{det}(Q) = -20 \][/tex]
- The determinant of matrix [tex]\( \frac{1}{2}P + Q \)[/tex] is:
[tex]\[ \text{det}\left(\frac{1}{2}P + Q\right) = -18.5 \][/tex]
These are the required determinants.
We are given two matrices [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] and we need to find the determinant of the matrix [tex]\(\frac{1}{2}P + Q\)[/tex].
Matrix [tex]\( P \)[/tex] is:
[tex]\[ P = \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} \][/tex]
Matrix [tex]\( Q \)[/tex] is:
[tex]\[ Q = \begin{pmatrix} 2 & 4 \\ 6 & 2 \end{pmatrix} \][/tex]
First, let's compute the determinant of [tex]\( P \)[/tex].
### Step 1: Determinant of [tex]\( P \)[/tex]
For a 2x2 matrix [tex]\( P = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the determinant is given by:
[tex]\[ \text{det}(P) = ad - bc \][/tex]
Substituting the values from matrix [tex]\( P \)[/tex]:
[tex]\[ \text{det}(P) = (2 \cdot 1) - (0 \cdot 1) = 2 - 0 = 2 \][/tex]
So, the determinant of [tex]\( P \)[/tex] is:
[tex]\[ \text{det}(P) = 2 \][/tex]
### Step 2: Determinant of [tex]\( Q \)[/tex]
Similarly, we compute the determinant of [tex]\( Q \)[/tex].
For matrix [tex]\( Q = \begin{pmatrix} 2 & 4 \\ 6 & 2 \end{pmatrix} \)[/tex]:
[tex]\[ \text{det}(Q) = (2 \cdot 2) - (4 \cdot 6) = 4 - 24 = -20 \][/tex]
So, the determinant of [tex]\( Q \)[/tex] is:
[tex]\[ \text{det}(Q) = -20 \][/tex]
### Step 3: Calculate Matrix [tex]\(\frac{1}{2}P + Q\)[/tex]
First, we need to compute [tex]\(\frac{1}{2}P\)[/tex].
[tex]\[ \frac{1}{2}P = \frac{1}{2} \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \][/tex]
Now, add this result to matrix [tex]\( Q \)[/tex].
[tex]\[ \frac{1}{2}P + Q = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} + \begin{pmatrix} 2 & 4 \\ 6 & 2 \end{pmatrix} = \begin{pmatrix} 1 + 2 & 0 + 4 \\ \frac{1}{2} + 6 & \frac{1}{2} + 2 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 6.5 & 2.5 \end{pmatrix} \][/tex]
### Step 4: Determinant of [tex]\(\frac{1}{2}P + Q\)[/tex]
Now we compute the determinant of the new matrix [tex]\( \frac{1}{2}P + Q \)[/tex].
For matrix [tex]\( \begin{pmatrix} 3 & 4 \\ 6.5 & 2.5 \end{pmatrix} \)[/tex]:
[tex]\[ \text{det}\left(\frac{1}{2}P + Q\right) = (3 \cdot 2.5) - (4 \cdot 6.5) = 7.5 - 26 = -18.5 \][/tex]
### Final Results:
- The determinant of [tex]\( P \)[/tex] is:
[tex]\[ \text{det}(P) = 2 \][/tex]
- The determinant of [tex]\( Q \)[/tex] is:
[tex]\[ \text{det}(Q) = -20 \][/tex]
- The determinant of matrix [tex]\( \frac{1}{2}P + Q \)[/tex] is:
[tex]\[ \text{det}\left(\frac{1}{2}P + Q\right) = -18.5 \][/tex]
These are the required determinants.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
