What does an inverse of a matrix produce when multiplied by the original matrix?

When an inverse of a matrix is multiplied by the original matrix, the result is the identity matrix. The identity matrix, similar to the number one in arithmetic, acts as the multiplicative identity in matrix operations. This means that for any square matrix ( A ), if ( A^{-1} ) is the inverse of ( A ), then the equation ( A \times A^{-1} = I ) holds true, where ( I ) is the identity matrix.

The identity matrix has special properties: it maintains the dimensions of the multiplied matrices and serves as the neutral element in multiplication, not changing other matrices when they are multiplied by it. This property is analogous to multiplying a number by one, which leaves the number unchanged.

The other options do not represent the outcome of multiplying a matrix by its inverse. A null matrix would mean all entries are zero, which does not occur in this case. The zero matrix and a scaled version of the original matrix also do not describe the result of the multiplication of a matrix by its inverse. Therefore, the identity matrix is the correct result of this multiplication, confirming the fundamental property of matrix inverses.