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Sagot :
Certainly! Let's verify the given statement: [tex]\((PQ)^{-1} = Q^{-1} P^{-1}\)[/tex] where the matrices [tex]\(P\)[/tex] and [tex]\(Q\)[/tex] are defined as follows:
[tex]\[ P = \begin{pmatrix} 4 & 5 \\ 7 & 9 \end{pmatrix} \][/tex]
[tex]\[ Q = \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \][/tex]
### Step 1: Calculate the product [tex]\(PQ\)[/tex]
First, we compute the matrix product [tex]\(PQ\)[/tex]:
[tex]\[ PQ = \begin{pmatrix} 4 & 5 \\ 7 & 9 \end{pmatrix} \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \][/tex]
We can find each element of the resulting matrix [tex]\(PQ\)[/tex] as follows:
- The element in the first row, first column is [tex]\(4 \cdot 6 + 5 \cdot 8 = 24 + 40 = 64\)[/tex]
- The element in the first row, second column is [tex]\(4 \cdot 7 + 5 \cdot 9 = 28 + 45 = 73\)[/tex]
- The element in the second row, first column is [tex]\(7 \cdot 6 + 9 \cdot 8 = 42 + 72 = 114\)[/tex]
- The element in the second row, second column is [tex]\(7 \cdot 7 + 9 \cdot 9 = 49 + 81 = 130\)[/tex]
Thus,
[tex]\[ PQ = \begin{pmatrix} 64 & 73 \\ 114 & 130 \end{pmatrix} \][/tex]
### Step 2: Calculate the inverse of [tex]\(P\)[/tex]
Next, we compute the inverse of matrix [tex]\(P\)[/tex]. We use the formula for the inverse of a [tex]\(2 \times 2\)[/tex] matrix:
[tex]\[ P^{-1} = \frac{1}{\text{det}(P)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \quad \text{where} \quad P = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
For [tex]\(P\)[/tex]:
[tex]\[ P = \begin{pmatrix} 4 & 5 \\ 7 & 9 \end{pmatrix} \][/tex]
The determinant, [tex]\(\text{det}(P)\)[/tex], is given by:
[tex]\[ \text{det}(P) = 4 \cdot 9 - 5 \cdot 7 = 36 - 35 = 1 \][/tex]
Hence,
[tex]\[ P^{-1} = \begin{pmatrix} 9 & -5 \\ -7 & 4 \end{pmatrix} \][/tex]
### Step 3: Calculate the inverse of [tex]\(Q\)[/tex]
Similarly, we calculate the inverse of matrix [tex]\(Q\)[/tex]:
[tex]\[ Q = \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \][/tex]
The determinant, [tex]\(\text{det}(Q)\)[/tex], is given by:
[tex]\[ \text{det}(Q) = 6 \cdot 9 - 7 \cdot 8 = 54 - 56 = -2 \][/tex]
Thus,
[tex]\[ Q^{-1} = \frac{1}{-2} \begin{pmatrix} 9 & -7 \\ -8 & 6 \end{pmatrix} = \begin{pmatrix} -4.5 & 3.5 \\ 4 & -3 \end{pmatrix} \][/tex]
### Step 4: Calculate the product [tex]\(Q^{-1} P^{-1}\)[/tex]
We now find the product of the inverses [tex]\(Q^{-1} P^{-1}\)[/tex]:
[tex]\[ Q^{-1} P^{-1} = \begin{pmatrix} -4.5 & 3.5 \\ 4 & -3 \end{pmatrix} \begin{pmatrix} 9 & -5 \\ -7 & 4 \end{pmatrix} \][/tex]
We calculate each element of the resulting matrix:
- The element in the first row, first column is [tex]\((-4.5 \cdot 9) + (3.5 \cdot -7) = -40.5 - 24.5 = -65\)[/tex]
- The element in the first row, second column is [tex]\((-4.5 \cdot -5) + (3.5 \cdot 4) = 22.5 + 14 = 36.5\)[/tex]
- The element in the second row, first column is [tex]\((4 \cdot 9) + (-3 \cdot -7) = 36 + 21 = 57\)[/tex]
- The element in the second row, second column is [tex]\((4 \cdot -5) + (-3 \cdot 4) = -20 - 12 = -32\)[/tex]
Thus,
[tex]\[ Q^{-1} P^{-1} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
### Step 5: Calculate the inverse of [tex]\(PQ\)[/tex]
Finally, we calculate the inverse of the product [tex]\(PQ \)[/tex]:
[tex]\[ PQ = \begin{pmatrix} 64 & 73 \\ 114 & 130 \end{pmatrix} \][/tex]
The determinant, [tex]\(\text{det}(PQ)\)[/tex], is:
[tex]\[ \text{det}(PQ) = 64 \cdot 130 - 73 \cdot 114 = 8320 - 8322 = -2 \][/tex]
Thus,
[tex]\[ (PQ)^{-1} = \frac{1}{-2} \begin{pmatrix} 130 & -73 \\ -114 & 64 \end{pmatrix} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
### Conclusion
Since:
[tex]\[ (PQ)^{-1} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
and
[tex]\[ Q^{-1} P^{-1} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
We have verified that:
[tex]\[ (PQ)^{-1} = Q^{-1} P^{-1} \][/tex]
This completes the solution!
[tex]\[ P = \begin{pmatrix} 4 & 5 \\ 7 & 9 \end{pmatrix} \][/tex]
[tex]\[ Q = \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \][/tex]
### Step 1: Calculate the product [tex]\(PQ\)[/tex]
First, we compute the matrix product [tex]\(PQ\)[/tex]:
[tex]\[ PQ = \begin{pmatrix} 4 & 5 \\ 7 & 9 \end{pmatrix} \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \][/tex]
We can find each element of the resulting matrix [tex]\(PQ\)[/tex] as follows:
- The element in the first row, first column is [tex]\(4 \cdot 6 + 5 \cdot 8 = 24 + 40 = 64\)[/tex]
- The element in the first row, second column is [tex]\(4 \cdot 7 + 5 \cdot 9 = 28 + 45 = 73\)[/tex]
- The element in the second row, first column is [tex]\(7 \cdot 6 + 9 \cdot 8 = 42 + 72 = 114\)[/tex]
- The element in the second row, second column is [tex]\(7 \cdot 7 + 9 \cdot 9 = 49 + 81 = 130\)[/tex]
Thus,
[tex]\[ PQ = \begin{pmatrix} 64 & 73 \\ 114 & 130 \end{pmatrix} \][/tex]
### Step 2: Calculate the inverse of [tex]\(P\)[/tex]
Next, we compute the inverse of matrix [tex]\(P\)[/tex]. We use the formula for the inverse of a [tex]\(2 \times 2\)[/tex] matrix:
[tex]\[ P^{-1} = \frac{1}{\text{det}(P)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \quad \text{where} \quad P = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
For [tex]\(P\)[/tex]:
[tex]\[ P = \begin{pmatrix} 4 & 5 \\ 7 & 9 \end{pmatrix} \][/tex]
The determinant, [tex]\(\text{det}(P)\)[/tex], is given by:
[tex]\[ \text{det}(P) = 4 \cdot 9 - 5 \cdot 7 = 36 - 35 = 1 \][/tex]
Hence,
[tex]\[ P^{-1} = \begin{pmatrix} 9 & -5 \\ -7 & 4 \end{pmatrix} \][/tex]
### Step 3: Calculate the inverse of [tex]\(Q\)[/tex]
Similarly, we calculate the inverse of matrix [tex]\(Q\)[/tex]:
[tex]\[ Q = \begin{pmatrix} 6 & 7 \\ 8 & 9 \end{pmatrix} \][/tex]
The determinant, [tex]\(\text{det}(Q)\)[/tex], is given by:
[tex]\[ \text{det}(Q) = 6 \cdot 9 - 7 \cdot 8 = 54 - 56 = -2 \][/tex]
Thus,
[tex]\[ Q^{-1} = \frac{1}{-2} \begin{pmatrix} 9 & -7 \\ -8 & 6 \end{pmatrix} = \begin{pmatrix} -4.5 & 3.5 \\ 4 & -3 \end{pmatrix} \][/tex]
### Step 4: Calculate the product [tex]\(Q^{-1} P^{-1}\)[/tex]
We now find the product of the inverses [tex]\(Q^{-1} P^{-1}\)[/tex]:
[tex]\[ Q^{-1} P^{-1} = \begin{pmatrix} -4.5 & 3.5 \\ 4 & -3 \end{pmatrix} \begin{pmatrix} 9 & -5 \\ -7 & 4 \end{pmatrix} \][/tex]
We calculate each element of the resulting matrix:
- The element in the first row, first column is [tex]\((-4.5 \cdot 9) + (3.5 \cdot -7) = -40.5 - 24.5 = -65\)[/tex]
- The element in the first row, second column is [tex]\((-4.5 \cdot -5) + (3.5 \cdot 4) = 22.5 + 14 = 36.5\)[/tex]
- The element in the second row, first column is [tex]\((4 \cdot 9) + (-3 \cdot -7) = 36 + 21 = 57\)[/tex]
- The element in the second row, second column is [tex]\((4 \cdot -5) + (-3 \cdot 4) = -20 - 12 = -32\)[/tex]
Thus,
[tex]\[ Q^{-1} P^{-1} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
### Step 5: Calculate the inverse of [tex]\(PQ\)[/tex]
Finally, we calculate the inverse of the product [tex]\(PQ \)[/tex]:
[tex]\[ PQ = \begin{pmatrix} 64 & 73 \\ 114 & 130 \end{pmatrix} \][/tex]
The determinant, [tex]\(\text{det}(PQ)\)[/tex], is:
[tex]\[ \text{det}(PQ) = 64 \cdot 130 - 73 \cdot 114 = 8320 - 8322 = -2 \][/tex]
Thus,
[tex]\[ (PQ)^{-1} = \frac{1}{-2} \begin{pmatrix} 130 & -73 \\ -114 & 64 \end{pmatrix} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
### Conclusion
Since:
[tex]\[ (PQ)^{-1} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
and
[tex]\[ Q^{-1} P^{-1} = \begin{pmatrix} -65 & 36.5 \\ 57 & -32 \end{pmatrix} \][/tex]
We have verified that:
[tex]\[ (PQ)^{-1} = Q^{-1} P^{-1} \][/tex]
This completes the solution!
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