Identity Matrix

A scalar matrix whose diagonal elements are all equal to 1, the identity element of the ground field F, is said to be an identity (or unit) matrix. The identity matrix of order n is denoted by In.

Thus In = \(\begin{bmatrix} 1 & 0 & ... & 0\\ 0 & 1 & ... & 0\\ ... & ... & ... & ...\\ 0 & 0 & ... & 1 \end{bmatrix}\) = (δij)m,n where δij = 1 if i = j,

                                                                          δij = 0 if i ≠ j.

A scalar matrix is said to be a unit matrix, if diagonal elements are unity.

\(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\) is a unit matrix.

It is generally represented by I

For example:

1. \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\) is an unit matrix of order 2.

2. \(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\) is an unit matrix of order 3.

3. \(\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}\) is an unit matrix of order 4.


I. If A is a square matrix of order n and I is a unit matrix of the same order then AI = IA = A.

For example:

Let, A = \(\begin{bmatrix} 3 & 4 & 5\\ 2 & 3 & 1\\ 6 & 7 & 3 \end{bmatrix}\),                                         I = \(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Then, AI = \(\begin{bmatrix} 3\cdot 1 + 4\cdot 0 + 5\cdot 0 & 3\cdot 0 + 4\cdot 1 + 5\cdot 0 & 3\cdot 0 + 4\cdot 0 + 5\cdot 1\\ 2\cdot 1 + 3\cdot 0 + 1\cdot 0 & 2\cdot 0 + 3\cdot 1 + 1\cdot 0 & 2\cdot 0 + 3\cdot 0 + 1\cdot 1\\ 6\cdot 1 + 7\cdot 0 + 3\cdot 0 & 6\cdot 0 + 7\cdot 1 + 3\cdot 0 & 6\cdot 0 + 7\cdot 0 + 3\cdot 1 \end{bmatrix}\)

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

= A

Similarly, IA = A


II. If [d] is a scalar matrix then [d] = dI

For example;

[2] = \(\begin{bmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2 \end{bmatrix}\)                                 I = \(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Now, 2I = \(\begin{bmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2 \end{bmatrix}\) = [2]

`





10th Grade Math

From Identity Matrix to HOME PAGE

New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.