Identity matrix

Definition

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning any matrix multiplied by an identity matrix remains unchanged.

5 Must Know Facts For Your Next Test

1. The identity matrix is always square, meaning it has the same number of rows and columns.
2. In notation, the identity matrix of size $n \times n$ is often denoted as $I_n$ or simply $I$ if the size is clear from context.
3. For any matrix $A$ of compatible dimensions, multiplying by an identity matrix satisfies $AI = IA = A$.
4. The inverse of a matrix $A$, if it exists, can be found such that $AA^{-1} = A^{-1}A = I$.
5. Identity matrices play a crucial role in solving systems of linear equations using inverse matrices.

Review Questions

• What properties make the identity matrix unique in terms of its elements?
• How does multiplying a matrix by an identity matrix affect the original matrix?
• Why is the concept of an identity matrix important when discussing inverses?

Related terms

Inverse Matrix: A matrix that, when multiplied by its original matrix, yields the identity matrix.

Matrix Multiplication: An operation that takes two matrices and produces another matrix by computing the dot product of rows and columns.

$n \times n$ Matrix: A square matrix with equal numbers of rows and columns, typically denoted by its dimensions such as $3 \times 3$, $4 \times 4$, etc.