Find the inverse of the matrix $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$.

Q: Find the inverse of the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

To find the inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, we use the formula: $$A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ **Step 1: Find the determinant $|A|$** $$|A| = (1)(4) - (2)(3) = 4 - 6 = -2$$ Since $|A| \neq 0$, the inverse exists. **Step 2: Find the adjugate matrix** Swap the diagonal elements and change the signs of the off-diagonal elements: $$\text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$| **Step 3: Calculate $A^{-1}$** $$A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$$

Question

Accepted Answer

To find the inverse of a $2 	imes 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, we use the formula:
$$A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$$

**Step 1: Find the determinant $|A|$**
$$|A| = (1)(4) - (2)(3) = 4 - 6 = -2$$
Since $|A| 
eq 0$, the inverse exists.

**Step 2: Find the adjugate matrix**
Swap the diagonal elements and change the signs of the off-diagonal elements:
$$	ext{adj}(A) = \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix}$|

**Step 3: Calculate $A^{-1}$**
$$A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \ 1.5 & -0.5 \end{pmatrix}$$

Find the inverse of the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$. - DDCET Q&A

Find the inverse of the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

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