2 By 2 Identity Matrix

Identity Matrix is the matrix which is northward× n square matrix where the diagonal consist of ones and the other elements are all zeros. Information technology is also called equally a Unit Matrix or Uncomplicated matrix. It is represented as I_n or just by I, where n represents the size of the square matrix. For case,

\(\begin{array}{50}I_{one}=1\\I_{two}=\begin{bmatrix} 1 &0 \\ 0 &one \terminate{bmatrix}\\ I_{3}=\begin{bmatrix} 1 & 0 & 0\\ 0 & one & 0 \\ 0 & 0 & 1 \stop{bmatrix}\terminate{assortment} \)

We can too say, the identity matrix is a type of diagonal matrix, where the chief diagonal elements are ones, and rest elements are zeros. Let's study near its definition, properties and practice some examples on it.

Identity Matrix Definition

An identity matrix is a foursquare matrix in which all the elements of principal diagonals are 1, and all other elements are zeros. It is denoted by the notation "I_n" or merely "I". If whatever matrix is multiplied with the identity matrix, the effect will be given matrix. The elements of the given matrix remain unchanged. In other words, if all the chief diagonal of a square matrix are one's and rest all o'southward, information technology is called an identity matrix. Hither, the 2 × 2 and 3 × 3 identity matrix is given beneath:

2 × 2 Identity Matrix

2x2 Identity Matrix

This is too chosen the identity matrix of social club 2.

three× 3 Identity Matrix

3X3 Identity Matrix

This is known as the identity matrix of order 3 or unit matrix of order 3 × three.

Identity Matrix is donated by I_{northward × n,}where n × n shows the order of the matrix.

A × I _{n × n} = A, A = any square matrix of order n × northward.

Too, read:

Changed Matrix
Orthogonal Matrix
Singular Matrix
Symmetric Matrix
Upper Triangular Matrix

Properties of Identity Matrix

i) It is always a Square Matrix

These Matrices are said to be foursquare equally it always has the same number of rows and columns. For whatever whole number n, there'southward a corresponding Identity matrix, north × due north.

2) By multiplying any matrix by the unit matrix, gives the matrix itself.

As the multiplication is not always defined, and then the size of the matrix matters when we work on matrix multiplication.

Similar, for "thou × n" matrix C, we get

I_mC = C = CI_n

Then the size of the matrix is important as multiplying past the unit is similar doing it past 1 with numbers. For example:

C =

\(\begin{assortment}{50}\begin{bmatrix} 1 & 2 & three &4 \\ v& 6& 7 & 8 \terminate{bmatrix}\stop{array} \)

The to a higher place is two × four matrix as it has two rows and 4 columns.

Allow's multiply the two × 2 identity matrix past C.

\(\begin{array}{l}\brainstorm{bmatrix} 1 & 0\\ 0 & 1 \cease{bmatrix}.\begin{bmatrix} 1 & ii & 3 &iv \\ 5& 6& 7 & 8 \terminate{bmatrix}=\begin{bmatrix} ane+0 & 2+0 & iii+0 &4+0 \\ 0+5& 0+half-dozen& 0+vii & 0+8 \end{bmatrix} = \begin{bmatrix} 1 & 2 & three &4 \\ 5& half dozen& seven & 8 \end{bmatrix}\terminate{assortment} \)

Hence proved.

three) We ever become an identity after multiplying 2 inverse matrices.

If we multiply ii matrices which are inverses of each other, then we get an identity matrix.

C =

\(\begin{assortment}{l}\begin{bmatrix} 0 &1 \\ -2& ane \end{bmatrix}\end{assortment} \)

\(\begin{array}{l}\brainstorm{bmatrix} \frac{1}{ii} &- \frac{1}{2} \\ one& 0 \end{bmatrix}\cease{assortment} \)

CD=

\(\brainstorm{array}{l}\begin{bmatrix} 0 &1 \\ -2& 1 \finish{bmatrix}\end{assortment} \)

\(\begin{array}{50}\begin{bmatrix} \frac{ane}{two} &- \frac{1}{2} \\ 1& 0 \finish{bmatrix}\end{array} \)

\(\begin{assortment}{fifty}\begin{bmatrix} one & 0\\ 0 & 1 \end{bmatrix}\end{assortment} \)

DC =

\(\begin{assortment}{l}\brainstorm{bmatrix} \frac{1}{2} &- \frac{1}{2} \\ 1& 0 \end{bmatrix}\cease{assortment} \)

\(\begin{array}{l}\begin{bmatrix} 0 &ane \\ -2& one \end{bmatrix}\end{array} \)

\(\begin{array}{l}\begin{bmatrix} i & 0\\ 0 & ane \end{bmatrix}\end{array} \)

Identity Matrix Examples

Example i: Write an instance of 4× 4 gild unit matrix.

Solution:The unit matrix is the one having ones on the primary diagonal & other entries as 'zeros'.

Thus, the unit of measurement matrix of society four × 4 or the identity matrix of society 4 can be written as:

\(\begin{array}{fifty}I_{4\times four}=\begin{bmatrix} i & 0 & 0 &0 \\ 0& 1 & 0 &0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 &ane \cease{bmatrix}\end{array} \)

Example 2: Check the post-obit matrix is Identity matrix?

\(\begin{assortment}{l}\begin{bmatrix} i & 0 & 0 &0 \\ 0& 1 & 0 &0 \\ 0 & 0 & one & 0\\ \terminate{bmatrix}\stop{assortment} \)

Solution: No, It'southward not an identity matrix, because it is of the order 3 × 4, which is not a foursquare matrix.

Instance 3: Check the following matrix is Identity matrix; B =

\(\begin{array}{50}\begin{bmatrix} 1 & 1 & 1\\ ane & one& one\\ 1 & 1 & 1 \finish{bmatrix}\end{array} \)

Solution: No, it is not a unit of measurement matrix as it doesn't contain the value of 0 beside one property of having diagonal values of 1.

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Ofttimes Asked Questions on Identity Matrix

What do you mean by an identity matrix?

In linear algebra, an identity matrix is a matrix of order nxn such that each master diagonal element is equal to one, and the remaining elements of the matrix are equal to 0.

What is the identity matrix of a 2×2?

An identity matrix of 2×two is a matrix with i's in the primary diagonal and zeros everywhere. The identity matrix of order two×2 is:
[1 0 0 ane].

What is the identity matrix of a iii×3?

An identity matrix of 3×3 is a matrix with ane's in the main diagonal and zeros everywhere. The identity matrix of society 3×3 is given by:
[1 0 0 0 i 0 0 0 1].

Is the identity matrix nonsingular?

Yes, the identity matrix is nonsingular since its determinant is not equal to 0. The identity matrix is the only idempotent matrix with a non-nil determinant. Therefore, we can also notice the inverse of the identity matrix.

How exercise yous create an identity matrix?

As nosotros know, the identity matrix has all its master diagonal elements as 1's and the remaining elements 0'south. Suppose to create an identity matrix of society 4×4, we write the matrix elements in rows and columns equally given beneath, and those should exist enclosed within [ ].
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 i