The Identity Matrix | Moosa Academy

The Identity Matrix — Moosa Academy

In ordinary multiplication the number 1 leaves everything unchanged. Matrices have their own version of that number, and it is called the identity matrix.

Concept The identity element

For ordinary numbers, multiplying by 1 changes nothing:

a \times 1 = 1 \times a = a

The identity matrix plays exactly this role for matrix multiplication. It is written $I$ , with a subscript for its size when that matters: $I_2$ , $I_3$ , and so on.

Concept What it looks like

Every entry on the main diagonal is 1; every other entry is 0. The matrix is always square, since only a square matrix has a full diagonal.

I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \qquad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

There is one identity matrix for every size — $I_2$ , $I_3$ , $I_4$ and so on upward.

Theorem The identity property

A \times I = I \times A = A

Multiplying in either order returns the original matrix — one of the rare cases where matrix multiplication does not care about order. The sizes must match: an $n \times n$ matrix needs $I_n$ .

Example Checking it entry by entry

Multiply $A = \begin{pmatrix} 3 & 5 \\ -2 & 7 \end{pmatrix}$ by $I_2$ .

Entry (1,1):

(3)(1) + (5)(0) = 3

Entry (1,2):

(3)(0) + (5)(1) = 5

Entry (2,1):

(-2)(1) + (7)(0) = -2

Entry (2,2):

(-2)(0) + (7)(1) = 7

⟹ the same matrix A

The zeros switch off every term except one, and the single 1 lets that term through untouched. That is the whole mechanism.

Summary

The identity element for numbers is 1; for matrices it is the identity matrix I.
I is square, with 1s on the main diagonal and 0s everywhere else.
There is one for each size: I₂, I₃, I₄ and so on.
A × I = I × A = A, in either order.
The two matrices must be the same size for the multiplication to work.