Among matrices there is a "do nothing" — the identity matrix I. It has 1s down the main diagonal and 0s everywhere else:

Multiply any vector by I and it comes back unchanged, I\vec{x} = \vec{x}; multiply any matrix by I and you get that matrix back, IA = AI = A. The identity plays the role for matrices that 1 plays for numbers: the neutral element of multiplication.

Nothing moves

The columns of I are exactly the standard unit vectors \hat{\imath} and \hat{\jmath} — so, read as a weighted sum of columns, I\vec{x} just rebuilds \vec{x}. Move the input below; the output arrow sits exactly on top of it, every time.

Why a "do nothing" is so useful

Because it's the goal of undoing. The inverse of a matrix A is the matrix A^{-1} that cancels it: A^{-1}A = I. "Get back to where you started" is the identity. There's one identity matrix for each size — I_2, I_3, and so on — always square, always 1s on the diagonal.