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Determinant of a 2x2 matrix

Determinants are useful for solving systems of linear equations; however, in the context of a 3D physics engine, we use them almost exclusively to find the inverse of a matrix. The determinant of a matrix M is a scalar value, it's denoted as Determinant of a 2x2 matrix .The determinant of a matrix is the same as the determinant of its transpose .

We can use a shortcut to find the determinant of a 2 X 2 matrix; subtract the product of the diagonals. This is actually the manually expanded form of Laplace Expansion; we will cover the proper formula in detail later:

One interesting property of determinants is that the determinant of the inverse of a matrix is the same as the inverse determinant of that matrix:

Finding the determinant of a 2 X 2 matrix is fairly straightforward, as we have already expanded the formula. We're just going to implement this in code.

How to do it…

Follow these steps to implement a function which returns the determinant of a 2 X 2 matrix:

Add the declaration for the determinant function to matrices.h:
```
float Determinant(const mat2& matrix);
```

Add the implementation for the determinant function to matrices.cpp:

float Determinant(const mat2& matrix) {
    return matrix._11 * matrix._22 –
           matrix._12 * matrix._21;
}

How it works…

Every square matrix has a determinant. We can use the determinant to figure out whether a matrix has an inverse or not. If the determinant of a matrix is non-zero, then the matrix has an inverse. If the determinant of a matrix is zero, then the matrix has no inverse.