Identity Matrix

Identity Matrix is also called as a Unit Matrix or Elementary matrix. Let’s study about its definition, Properties and practice some examples on it.

Definition of Identity Matrix

If all the main diagonal of a square matrix are 1’s and rest all o’s, it is called an identity matrix.

Example of Identity Matrix

2 x 2 Identity Matrix

Identity Matrix 2x2

3 x 3 Identity Matrix

3X3 Identity Matrix

It is a type of matrix which when multiplied by another matrix gives equal to that matrix.

Identity Matrix 2X2

Identity Matrix is donated by In X n , where n X n shows the order of the matrix.

A X I n X n = A, A = any square matrix of order n X n.

Problems on Identity Matrix

Example 1: Write an example of 4 X 4 order unit matrix.

Solution:

The unit matrix is the one having ones on the main diagonal & other entries as ‘zeros’.

Example 2: Check the following matrix is Identity matrix?

V= \(\begin{bmatrix} 1 & 0 & 0 &0 \\ 0& 1 & 0 &0 \\ 0 & 0 & 1 & 0\\ \end{bmatrix}\)

Solution: No, It’s not an identity matrix, because it is of the order 3 X 4, which is not a square matrix.

Example 3:

B = \(\begin{bmatrix} 1 & 1 & 1\\ 1 & 1& 1\\ 1 & 1 & 1 \end{bmatrix}\)

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

Properties of Identity Matrix

It is always a Square Matrix

These Matrices are said to be square as it always has the same number of rows and columns. For any whole number n, there’s a corresponding Identity matrix, n x n.

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

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

Like, for m x n matrix C, we get

ImC = C = CIn

So the size of the matrix is important as multiplying by the unit is like doing it by 1 with numbers. For example:

C = \(\begin{bmatrix} 1 & 2 & 3 &4 \\ 5& 6& 7 & 8 \end{bmatrix}\)

The above is 2 x 4 matrix as it has 2 rows and 4 columns.

3) We always get an identity after multiplying two inverse matrices.

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

C = \(\begin{bmatrix} 0 &1 \\ -2& 1 \end{bmatrix}\)

D= \(\begin{bmatrix} \frac{1}{2} &- \frac{1}{2} \\ 1& 0 \end{bmatrix}\)

CD= \(\begin{bmatrix} 0 &1 \\ -2& 1 \end{bmatrix}\)\(\begin{bmatrix} \frac{1}{2} &- \frac{1}{2} \\ 1& 0 \end{bmatrix}\) = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\)

DA = \(\begin{bmatrix} \frac{1}{2} &- \frac{1}{2} \\ 1& 0 \end{bmatrix}\) \(\begin{bmatrix} 0 &1 \\ -2& 1 \end{bmatrix}\) = \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\)

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Practise This Question

If the vectors a,b and c form the sides BC, CA and AB respectively of a Δ ABC, then

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