Solved Examples on Identity Matrix

Example 1: Determine whether the given matrices are identity matrices or not.

a) A = [Tex]\left[\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right][/Tex]

b) B= [Tex]\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right][/Tex]

Solution:

a) In matrix A, all the principal diagonal elements are ones, and the rest of the elements are also equal to 1. We know that a unit matrix, or identity matrix, is a square matrix whose all elements are zeros except the main diagonal elements, which are ones. Hence, matrix A is not an identity matrix.

b) In matrix B, all the principal diagonal elements are ones, and the rest of the elements are zeros. So, from the definition of an identity matrix, the given matrix B is an identity matrix.

Example 2: Give an example of an identity matrix that has four rows and four columns.

Solution:

The order of an identity matrix that has four rows and four columns is “4 × 4.” The matrix given below represents an identity matrix of order “4 × 4,” where all the principal diagonal elements are ones, and the rest of the elements are zeros.

I_4×4 = [Tex]\left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right][/Tex]

Example 3: Find the value of (p − q + r) if the matrix given below is an identity matrix.

Solution:

If the given matrix is an identity matrix, then all its principal diagonal elements are ones, and the rest of the elements are zeros.

So, p = 1

q + 1 = 0 ⇒ q = -1

r – 2 = 0 ⇒ r = 2

Now, p − q + r = 1 −(−1) + 2

= 1 + 1 + 2 = 4

Hence, the value of (p − q + r) is 4 if matrix A is an identity matrix.

Example 4: Prove that the inverse of the identity matrix is the identity matrix itself.

Solution:

Let’s consider an identity matrix of order “2 × 2” to prove that the inverse of the identity matrix is the identity matrix itself.

I_{2 × 2} = [Tex]\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right][/Tex]

We know that the inverse of a matrix A [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]   [/Tex] = Adj A/ (ad – bc)

where Adj A = [Tex]\left[\begin{array}{cc} d & -b\\ -c & a \end{array}\right][/Tex]

I^-1 = 1/ (1 – 0) [Tex]\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right][/Tex]

I^-1 = [Tex]\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]   [/Tex] = I

Hence proved.

Example 5: For a given matrix A, prove that AI = A.

Solution:

Let’s consider a square matrix A = [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]   [/Tex] to prove that AI = A.

AI = [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right] [/Tex] [Tex]\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]   [/Tex] 

=  [Tex]\left[\begin{array}{cc} (a\times1+b\times0) & (0\times a+b\times1)\\ (c\times1+d\times0) & (0\times c+d\times1) \end{array}\right][/Tex]

= [Tex]\left[\begin{array}{cc} (a+0) & (0+b)\\ (c+0) & (0+d) \end{array}\right][/Tex]

= [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]   [/Tex] = A

Hence, proved.

Identity Matrix is a square matrix whose all diagonal elements are equal to 1 and rest all elements are zero. Identity Matrix is also known as Unit Matrix. In simple terms, all diagonal elements are equal to 1 and rest are zero. The main condition for Identity matrix is that it should be a square matrix i.e. the no of rows should be equal to zero.