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. It works by first finding the determinant, then swapping the main diagonal entries, changing the signs of the off-diagonal entries, and multiplying by the reciprocal of the determinant. This makes it useful for checking answers, understanding how a matrix inverse is found, and practising linear algebra step by step.
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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