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 I_n.

Thus I_n = \(\begin{bmatrix} 1 & 0 & ... & 0\\ 0 & 1 & ... & 0\\ ... & ... & ... & ...\\ 0 & 0 & ... & 1 \end{bmatrix}\) = (δ_ij)_m,n where δ_ij = 1 if i = j,

                                                                          δ_ij = 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

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