askvity

What is the determinant of the identity matrix?

Published in Linear Algebra 2 mins read

The determinant of the identity matrix is 1.

Let's delve into why this is the case:

Understanding the Identity Matrix

The identity matrix, denoted as I, is a square matrix where all the elements on the main diagonal are 1, and all other elements are 0. For example, a 3x3 identity matrix looks like this:

| 1  0  0 |
| 0  1  0 |
| 0  0  1 |

Calculating the Determinant

The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix.

For a 2x2 matrix:

| a  b |
| c  d |

The determinant is calculated as: ad - bc

For a 3x3 matrix:

| a  b  c |
| d  e  f |
| g  h  i |

The determinant is calculated as: a(ei - fh) - b(di - fg) + c(dh - eg)

Identity Matrix Determinant

For the identity matrix, regardless of its size (2x2, 3x3, nxn), applying the determinant calculation always results in 1.

Example (2x2 Identity Matrix):

| 1  0 |
| 0  1 |

Determinant = (1 1) - (0 0) = 1

Example (3x3 Identity Matrix):

| 1  0  0 |
| 0  1  0 |
| 0  0  1 |

Determinant = 1(11 - 00) - 0(01 - 00) + 0(00 - 10) = 1(1) - 0 + 0 = 1

Key Properties and Reasoning

  • The determinant of a matrix represents the scaling factor of the transformation represented by that matrix.
  • The identity matrix represents a transformation that doesn't change anything (it's the "do nothing" transformation). Therefore, the scaling factor is 1.
  • The determinant of a diagonal matrix (a matrix where all non-diagonal elements are zero) is the product of its diagonal elements. In the case of the identity matrix, all diagonal elements are 1, so the product is always 1.

Therefore, the determinant of the identity matrix is always 1, indicating that the identity matrix performs no scaling or transformation of space.

Related Articles