A worked example
Multiplying [[1,2],[3,4]] by [[5,6],[7,8]] gives [[19,22],[43,50]].
Frequently asked questions
Why does matrix multiplication need matching dimensions?
Each entry in the result comes from pairing up a full row from the first matrix with a full column from the second — if the row length and column length don't match, there's nothing valid to pair against.
What does it mean if a matrix has no inverse?
A matrix is only invertible if its determinant is nonzero — a zero determinant means the matrix 'collapses' space in a way that can't be undone, similar to how you can't divide by zero in regular arithmetic.