What is Identity Matrix and its Examples

what is Identity matrix?

An Identity matrix is a matrix in which the diagonals entries are 1 and off diagonals entries are zero. An Identity matrix is always a square matrix.

Identity Matrix Order
Identity Matrix Order

As we know that square matrix is in which a number of rows is equal to the number of columns. The identity matrix is denoted by I_{n\times n} where n by n shows the order of the identity matrix.it plays an important role in linear algebra since.

  • If we multiply any real number with 1 then its remains unchanged.
  • If we multiply any matrix with identity matrix then its remains unchanged.

General form of identity matrix

Identity Matrix Examples

Single matrix

I =\begin{bmatrix}1\end{bmatrix}

The order of the above matrix is 2 by 2

i_2 =\begin{bmatrix}1&0\\0&1\end{bmatrix}

The order of this matrix is 4 by 4

i_4 =\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}

The order of the above matrix is n by n

i_n =\begin{bmatrix}1&0&0&…&0\\0&1&0&…&0\\0&0&1&…&0\\…&…&…&…&…\\0&0&0&…&1\end{bmatrix}

Notations of Identity Matrix

The term unit matrix is widely used at an early age but now the term identity matrix has become a standard form. There is more notation used in different books such that in group theory the identity matrix is sometimes denoted by boldface one like 1. Some mathematics books use U or E where U stands for unit matrix and E stand for a germen word Einheitsmatrix.

 There is one more notation for identity matrix is KRONECKER DELTA.

If we have a matrix A with order m by n then it is a property of multiplication is that

l_mA= AI_n= A.

properties of identity matrix

  1. If we multiplying the two inverse matrices then we always get identity matrix in result such that
  2. If we multiply any matrix with identity matrix the result is identity matrix

I_mC = C = CI_n

  • The identity matrix is the only matrix which is idempotent matrix with non-zero determinant that means it is the only matrix which have
  • When the multiply by itself the result Is itself
  • All of rows and columns are linearly independent
  • The square root of  an identity matrix is itself and this is the only positive definite square root
  • It is always square matrix as the number of rows and the number of columns are equal

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