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Sagot :
To find the inverse of the matrix
[tex]\[ \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix}, \][/tex]
we follow these steps:
### Step 1: Determine the determinant
The determinant of the matrix
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
is calculated as
[tex]\[ \text{det} = ad - bc. \][/tex]
For our specific matrix:
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix}, \][/tex]
the determinant is:
[tex]\[ \text{det} = (2)(1) - (0)(9) = 2. \][/tex]
### Step 2: Check if the determinant is non-zero
Since the determinant (2) is non-zero, the matrix is invertible.
### Step 3: Find the adjugate of the matrix
The adjugate (or adjoint) of the matrix
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
is computed by swapping the elements on the main diagonal and changing the sign of the off-diagonal elements:
[tex]\[ \text{adj} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. \][/tex]
Applying this to our matrix:
[tex]\[ \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix} , \][/tex]
the adjugate matrix is:
[tex]\[ \begin{pmatrix} 1 & -0 \\ -9 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -9 & 2 \end{pmatrix}. \][/tex]
### Step 4: Calculate the inverse
The inverse of a matrix
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
is given by
[tex]\[ \text{inverse} = \frac{1}{\text{det}} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. \][/tex]
For our matrix, this becomes:
[tex]\[ \text{inverse} = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ -9 & 2 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ -\frac{9}{2} & 1 \end{pmatrix} = \begin{pmatrix} 0.5 & 0 \\ -4.5 & 1 \end{pmatrix}. \][/tex]
Thus, the inverse of the matrix
[tex]\[ \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix} \][/tex]
is
[tex]\[ \begin{pmatrix} 0.5 & 0 \\ -4.5 & 1 \end{pmatrix}. \][/tex]
### Step 5: Rounding (if necessary)
In this case, the matrix elements are already precise to the hundredth place, so no additional rounding is necessary.
Therefore, the inverse of the given matrix is
[tex]\[ \begin{pmatrix} 0.5 & 0 \\ -4.5 & 1 \end{pmatrix}. \][/tex]
[tex]\[ \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix}, \][/tex]
we follow these steps:
### Step 1: Determine the determinant
The determinant of the matrix
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
is calculated as
[tex]\[ \text{det} = ad - bc. \][/tex]
For our specific matrix:
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix}, \][/tex]
the determinant is:
[tex]\[ \text{det} = (2)(1) - (0)(9) = 2. \][/tex]
### Step 2: Check if the determinant is non-zero
Since the determinant (2) is non-zero, the matrix is invertible.
### Step 3: Find the adjugate of the matrix
The adjugate (or adjoint) of the matrix
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
is computed by swapping the elements on the main diagonal and changing the sign of the off-diagonal elements:
[tex]\[ \text{adj} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. \][/tex]
Applying this to our matrix:
[tex]\[ \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix} , \][/tex]
the adjugate matrix is:
[tex]\[ \begin{pmatrix} 1 & -0 \\ -9 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -9 & 2 \end{pmatrix}. \][/tex]
### Step 4: Calculate the inverse
The inverse of a matrix
[tex]\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \][/tex]
is given by
[tex]\[ \text{inverse} = \frac{1}{\text{det}} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. \][/tex]
For our matrix, this becomes:
[tex]\[ \text{inverse} = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ -9 & 2 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ -\frac{9}{2} & 1 \end{pmatrix} = \begin{pmatrix} 0.5 & 0 \\ -4.5 & 1 \end{pmatrix}. \][/tex]
Thus, the inverse of the matrix
[tex]\[ \begin{pmatrix} 2 & 0 \\ 9 & 1 \end{pmatrix} \][/tex]
is
[tex]\[ \begin{pmatrix} 0.5 & 0 \\ -4.5 & 1 \end{pmatrix}. \][/tex]
### Step 5: Rounding (if necessary)
In this case, the matrix elements are already precise to the hundredth place, so no additional rounding is necessary.
Therefore, the inverse of the given matrix is
[tex]\[ \begin{pmatrix} 0.5 & 0 \\ -4.5 & 1 \end{pmatrix}. \][/tex]
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