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How does the identity matrix behave in matrix multiplication?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, the identity matrix (denoted as I or I_n) acts as the multiplicative identity, functioning similarly to the number 1 in ordinary arithmetic.1

When a square matrix B is multiplied by an identity matrix of the same order, the result is always the original matrix B. This relationship is expressed by the equation:
$B I = I B = B$

Because the identity matrix leaves any matrix unchanged under multiplication, it serves as the neutral element in matrix algebra. Additionally, the identity matrix is essential in the concept of an invertible matrix, where a nonsingular matrix B multiplied by its inverse B⁻¹ yields the identity matrix (BB⁻¹ = B⁻¹B = I).12