The Identity Matrix
The Additive Identity
The identity property of addition states that when zero is added to any real number, the number does not change. Thus, the number "0" is called the additive identity for real numbers.
There is a matrix which is an additive identity for matrices:
This matrix is also denoted . When  is added to any matrix of the same dimensions, the matrix does not change. For example,
The Multiplicative Identity
The identity property of multiplication states that when 1 is multiplied by any real number, the number does not change; that is, any number times 1 is equal to itself. The number "1" is called the multiplicative identity for real numbers.
There is a matrix which is a multiplicative identity for matrices—the
This matrix, denoted I , is a square matrix. When any m×n matrix is multiplied on the left by an m×m identity matrix, or on the right by an n×n identity matrix, the m×n matrix does not change. For example,