Inverse of a 2 by 2 Matrix
In Different Ways
Let's calculate the Inverse of a simple 2×2 matrix in different ways!
Let's work on this simple guy:
Matrix Calculator
Using the Matrix Calculator we get "-2 1 1.5 -0.5 ":
Matrix Inverse by Row Operations
We can use Matrix Inverse by Row Operations.
Start with our matrix side-by-side with the Identity Matrix:
| A | I | |||
| 1 | 2 | 1 | 0 | |
| 3 | 4 | 0 | 1 | |
| 3 | 6 | 3 | 0 | times 3 |
| 3 | 4 | 0 | 1 | |
| 3 | 6 | 3 | 0 | |
| 0 | −2 | −3 | 1 | minus row 1 |
| 3 | 6 | 3 | 0 | |
| 0 | 6 | 9 | −3 | times −3 |
| 3 | 0 | -6 | 3 | minus row 2 |
| 0 | 6 | 9 | −3 | |
| 1 | 0 | −2 | 1 | times 1/3 |
| 0 | 1 | 1.5 | −0.5 | times 1/6 |
| I | A-1 | |||
And the result is:
Inverse of a 2×2 Matrix Shortcut
For a 2×2 Matrix like above there's a shortcut:
First find the determinant, which for a 2×2 Matrix is ad − bc:
= 4 − 6
= −2
Next, swap the positions of a and d, and change the signs of b and c:
Last, multiply every element by 1determinant:
Inverse of a Matrix using Minors, Cofactors and Adjugate
We can use Inverse of a Matrix using Minors, Cofactors and Adjugate
For a simple 2×2 matrix it is like using a Mercedes to crack a walnut, but let's have a go anyway.
The steps are:
- Step 1: calculating the Matrix of Minors,
- Step 2: then turn that into the Matrix of Cofactors,
- Step 3: then the Adjugate, and
- Step 4: multiply that by 1determinant
Step 1: Matrix of Minors
For each element of the matrix:
- ignore the values on the current row and column
- calculate the determinant of the remaining values
Put those determinants into a matrix (the "Matrix of Minors")
Example:
For the top left element, after ignoring its own row and column; only the bottom right remains.
So for a 2×2 matrix, it simplifies to just putting in the value diagonally opposite:
Step 2: Matrix of Cofactors
This is easy! Just apply a "checkerboard" of minuses to the "Matrix of Minors". In other words, we need to change the sign of alternate cells, like this:
Step 3: Adjugate (also called Adjoint)
Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same):
Step 4: Multiply by 1/Determinant
Now find the determinant of the original matrix. For a 2×2 matrix it is:
Now we multiply the Adjugate by 1determinant to get:
And we are done!
Conclusion
There are many ways to find the inverse of a matrix. And there are more ways than we have shown here. Hopefully they all lead to the same result!
- Using the Matrix Calculator is easiest
- Row Operations are fun, like solving a puzzle!
- The 2×2 Matrix Shortcut is best for a 2×2 Matrix
- Minors, Cofactors and Adjugate is way too much work for a 2×2, but useful for 3×3s and so on