Unit Matrix

Unit matrix is also called the identity matrix. Unit matrix is used as the multiplicative identity of square matrices in the matrices concept. When any square matrix is multiplied by the identity matrix, then the result doesn't change. In linear algebra, the unit matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

We use the unit matrix in proofs when determining the inverse of a matrix. In this article, let's learn about the unit matrix, its properties with solved examples.

The unit matrix is every n x n square matrix made up of all zeros except for the elements of the main diagonal that are all ones. When the unit matrix is the product of two square matrices A and B, then A and B are said to be the inverses of each other.

Unit Matrix Definition

A unit matrix can be defined as a scalar matrix in which all the diagonal elements are equal to 1 and all the other elements are zero. The unit matrix with the order n can be denoted by I_n. An identity matrix or unit matrix is generally represented by I. Let's see the example of unit matrices of orders 2 and 3.

We can see that in all the above-given matrices, the diagonal elements are all equal to 1 and the rest of the elements are 0. Hence, we call these matrices unit matrices.

A unit matrix can be considered as a diagonal matrix where all the diagonal elements are equal to 1. There are some important properties of the identity matrix that are used in linear algebra. The properties of the unit matrix are:

Property 1: If A is considered to be a square matrix of order n and I is considered to be a unit matrix of order n, then AI = IA = A. The identity functions like the number 1 (multiplicative identity for real numbers) in such a way that if you multiply a matrix by the unit matrix of the same order, you get the same initial matrix.

Let A = \(\left[\begin{array}{lll}
3 & 4 & 5 \\
2 & 3 & 8 \\
6 & 7 & 3
\end{array}\right]\)

and I = \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)

Then, A I = \(\left[\begin{array}{lll}
3 \cdot 1+4 \cdot 0+5 \cdot 0 & 3 \cdot 0+4 \cdot 1+5 \cdot 0 & 3 \cdot 0+4 \cdot 0+5 \cdot 1 \\
2 \cdot 1+3 \cdot 0+8 \cdot 0 & 2 \cdot 0+3 \cdot 1+8 \cdot 0 & 2 \cdot 0+3 \cdot 0+8 \cdot 1 \\
6 \cdot 1+7 \cdot 0+3 \cdot 0 & 6 \cdot 0+7 \cdot 1+3 \cdot 0 & 6 \cdot 0+7 \cdot 0+3 \cdot 1
\end{array}\right]\)

= \(\left[\begin{array}{lll}
3 & 4 & 5 \\
2 & 3 & 8 \\
6 & 7 & 3
\end{array}\right]\)

= A.

Similarly, it can be proved that IA = A.

Property 2: If [b] is a scalar matrix then [b] = bI.

Consider [3] = \(\left[\begin{array}{lll}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3
\end{array}\right]\)

and I = \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)

Then, 3I = \(\left[\begin{array}{lll}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3
\end{array}\right]\)

= [3]

Property 3: When one unit matrix of order n is multiplied by another unit matrix of same order, then the resultant matrix will also be a unit matrix of order n.

Property 4: When the inverse operation is applied on a unit matrix, the resultant matrix will be unit matrix itself. I^-1 = I.

Property 5: The determinant of a unit matrix is always equal to 1. det|I| = 1.

Important Notes on Unit Matrix

Here is a list of a few points that should be remembered while studying the unit matrix

• A unit matrix can be defined as a scalar matrix in which all the diagonal elements are equal to 1 and all the other elements are zero.
• Unit matrix is also called the identity matrix.
• Unit matrix is used as the multiplicative identity of square matrices in the matrices concept.

☛ Related Topics:

Check out the following pages related to the unit matrix

• Matrix Calculator
• Determinant of Matrix
• Diagonal Matrix Calculator
• Transpose Matrix Calculator

What is the Definition of a Unit Matrix?

A unit matrix can be defined as a scalar matrix in which all the diagonal elements are equal to 1 and all the other elements are zero. The unit matrix with the order n can be denoted by I_n. A unit or an identity matrix is generally represented by I.

What Is a Unit Matrix Example?

The below-given matrix is an example of a unit matrix. In matrix A, all the diagonal elements are equal to 1, and all the other elements are zero. Thus, A is a unit matrix.

A = \(\left[\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right]\)

Is Unit Matrix and Identity Matrix Same?

Yes, unit matrix and identity matrix are the same. Identity matrix is another term that is used to denote unit matrix.

Is Unit Matrix a Scalar Matrix?

Yes, a unit matrix can be considered as a scalar matrix in which all the diagonal elements are equal to 1 and the remaining elements are 0. A scalar matrix can be considered as a unit matrix that is multiplied with a scalar quantity 1.

What is the Rank of Unit Matrix of Order n?

The rank of a unit matrix of order n is always equal to n.

What Is a Unit Diagonal Matrix?

A unit matrix can be considered as a diagonal matrix where all the diagonal elements are considered to be unity. Unit matrix of order 'n' can be denoted by I_n or I.

How Do You Represent the Unit Matrix?

A unit matrix can be represented by I where all the diagonal elements will be equal to 1 and the remaining elements will be equal to 0.

What Is the Rank of a 3x3 Unit Matrix?

The rank of a 3×3 unit matrix is 3 as it has 3 linearly independent rows (or columns).