What the determinant tells you
Think of a matrix as a transformation that stretches, rotates or shears space. The determinant is the factor by which it scales area (for 2×2) or volume (for 3×3). A determinant of 3 means every shape comes out three times bigger. A negative determinant means the transformation also flips orientation, like a mirror.
A determinant of exactly zero means space gets squashed flat, onto a line or a plane, and there's no way to undo that. So the matrix has no inverse, and a set of simultaneous equations with those coefficients has either no solution or infinitely many.
For A = [[4, 7], [2, 6]], det(A) = 4 × 6 − 7 × 2 = 10.
Swap the diagonal entries, negate the other two, and divide by 10:
A⁻¹ = 1/10 × [[ 6, −7], = [[ 0.6, −0.7],
[−2, 4]] [−0.2, 0.4]]
Check: the top-left entry of A × A⁻¹ is 4 × 0.6 + 7 × (−0.2) = 2.4 − 1.4 = 1, as it should be.
The 3×3 method
For a 3×3 matrix, Kalkuli expands the determinant along the first row. Each entry is multiplied by the determinant of the 2×2 matrix left when you cross out its row and column (its minor), with signs alternating + − +. The inverse is the adjugate (the transpose of the matrix of signed minors, called cofactors) divided by the determinant.
This is exactly the method taught in A-level Further Maths and first-year engineering courses, so the working on the right should match what you'd write by hand. I've written out a full example, with every cofactor, in how to find the inverse of a 3×3 matrix by hand.
If the determinant is tiny compared with the entries, the inverse will have huge entries and be very sensitive to small changes in A. Kalkuli treats a determinant smaller than 10⁻¹² times the size of the entries (raised to the matrix order) as zero, because at that point the "inverse" would be mostly rounding error.
Trace and transpose
The trace is the sum of the main diagonal. It equals the sum of the eigenvalues, just as the determinant equals their product. The transpose swaps rows and columns; a matrix equal to its own transpose is called symmetric, and those turn up constantly in physics and statistics (stress tensors, covariance matrices).