Inverse of a 2×2 Matrix Calculator
This Inverse of a 2×2 Matrix Calculator helps you find the inverse of a 2×2 matrix and shows each step clearly. In Algebra 2, inverse matrices are often used when studying matrices, determinants, and systems of equations. The calculator first finds the determinant, then swaps the main diagonal entries, changes the signs of the off-diagonal entries, and multiplies by the reciprocal of the determinant.
Step-by-step method
- Write the matrix A.
- Use det( A ) = ad − bc.
- Substitute values to compute det( A ).
- Use A⁻¹ = ( 1 / det( A ) ) · [[d, −b], [−c, a]].
Determinant formula
a | b |
c | d |
Inverse formula
| 1 |
| det( A ) |
d | −b |
−c | a |
Example 1: 2×2 matrix inverse
Step 1 - Write the matrix A.
In this problem: We start with the given matrix A.
2 | 1 |
5 | 3 |
Step 2 - Use det( A ) = ad − bc.
In this problem: Use the determinant formula for a 2×2 matrix.
Step 3 - Substitute values to compute det( A ).
In this problem: Substitute values to get det( A ) = 1.
Step 4 - Use A⁻¹ = ( 1 / det( A ) ) · [[d, −b], [−c, a]].
In this problem: Now compute A⁻¹ using the inverse formula.
| 1 |
| det( A ) |
3 | -1 |
-5 | 2 |
3 | -1 |
-5 | 2 |
Final answer:
3 | -1 |
-5 | 2 |
Example 2: 2×2 matrix inverse
Step 1 - Write the matrix A.
In this problem: We start with the given matrix A.
4 | -2 |
1 | 1 |
Step 2 - Use det( A ) = ad − bc.
In this problem: Use the determinant formula for a 2×2 matrix.
Step 3 - Substitute values to compute det( A ).
In this problem: Substitute values to get det( A ) = 6.
Step 4 - Use A⁻¹ = ( 1 / det( A ) ) · [[d, −b], [−c, a]].
In this problem: Now compute A⁻¹ using the inverse formula.
| 1 |
| det( A ) |
1 | 2 |
-1 | 4 |
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Final answer:
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