Question:
If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A^{-1} =$
If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A^{-1} =$
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For $2 \times 2$ matrices, swap the main diagonal elements, change signs of off-diagonal elements, and divide by the determinant.
Updated On: Jun 3, 2026
- $-\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$
- $\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$
- $-\frac{1}{2} \begin{pmatrix} 1 & -2 \\ -3 & 4 \end{pmatrix}$
- $\frac{1}{2} \begin{pmatrix} 1 & -2 \\ -3 & 4 \end{pmatrix}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The inverse of a $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula: \[ M^{-1} = \frac{1}{\det(M)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
Step 2: Meaning
We compute the determinant of the matrix $A$ and swap the diagonal elements, while changing the sign of the off-diagonal elements.
Step 3: Analysis
First, calculate the determinant of $A$: \[ \det(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \] Now, find the adjoint of $A$: \[ \text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \] Therefore: \[ A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = -\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]
Step 4: Conclusion
The inverse matrix $A^{-1}$ is $-\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$.
Final Answer: (A)
The inverse of a $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula: \[ M^{-1} = \frac{1}{\det(M)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
Step 2: Meaning
We compute the determinant of the matrix $A$ and swap the diagonal elements, while changing the sign of the off-diagonal elements.
Step 3: Analysis
First, calculate the determinant of $A$: \[ \det(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \] Now, find the adjoint of $A$: \[ \text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \] Therefore: \[ A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = -\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]
Step 4: Conclusion
The inverse matrix $A^{-1}$ is $-\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$.
Final Answer: (A)
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