QuestionНайдите матрицы, обратные к след. a.) $\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$

Studdy Solution

STEP 1

1. The problem involves finding the inverse of a 2x2 matrix.
2. The inverse of a matrix $A$ is denoted as $A^{-1}$ and satisfies the equation $AA^{-1} = I$ , where $I$ is the identity matrix.
3. For a 2x2 matrix $A = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$ , the inverse is given by $A^{-1} = \frac{1}{\det(A)} \left( \begin{array}{cc} d & -b \\ -c & a \end{array} \right)$ , where $\det(A) = ad - bc$ is the determinant of $A$ .

STEP 2

1. Verify that the matrix is invertible by checking that the determinant is non-zero.
2. Calculate the determinant of the given matrix.
3. Use the formula for the inverse of a 2x2 matrix to find the inverse.

STEP 3

Verify that the matrix $\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$ is invertible by checking that its determinant is non-zero.
The determinant of the matrix $A$ is given by
$\det(A) = \left(2 \cdot 3\right) - \left(0 \cdot 0\right) = 6$
Since $\det(A) = 6 \neq 0$ , the matrix is invertible.

STEP 4

Calculate the determinant of the given matrix explicitly to confirm its non-zero value.
$\det\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right) = 2 \cdot 3 - 0 \cdot 0 = 6$

STEP 5

Use the formula for the inverse of a 2x2 matrix to find the inverse.
The formula for the inverse is
$A^{-1} = \frac{1}{\det(A)} \left( \begin{array}{cc} d & -b \\ -c & a \end{array} \right)$
For the matrix $A = \left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$ , we have $a = 2$ , $b = 0$ , $c = 0$ , $d = 3$ . Substituting these values in, we get
$A^{-1} = \frac{1}{6} \left( \begin{array}{cc} 3 & 0 \\ 0 & 2 \end{array} \right)$

STEP 6

Simplify the expression to obtain the final form of the inverse matrix.
$A^{-1} = \left( \begin{array}{cc} \frac{3}{6} & 0 \\ 0 & \frac{2}{6} \end{array} \right) = \left( \begin{array}{cc} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{array} \right)$
Solution: The inverse of the given matrix is
$\left( \begin{array}{cc} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{array} \right)$

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