🟩 Identity Matrix

In math, multiplying by certain values can leave an object unchanged. For example, multiplying by 1 keeps a number the same.

We can do this for matrices too — by multiplying with the constant 1. But is there a matrix equivalent of multiplying by 1? A matrix that doesn’t change the matrix it multiplies?

Let’s call this the identity matrix, since it preserves the identity of whatever it multiplies.

🎯 Initial Idea

Matrix multiplication is read: Row-wise from the left matrix Column-wise from the right matrix

Then we apply a sum of paired products.

To preserve the original matrix, we need a way to select each element without altering it.

🧱 Starting Conditions

Since the identity matrix is meant to behave like a constant, it’s natural to first place it on the right side of the multiplication.

📐 Shape Compatibility

For matrix multiplication to work, The identity matrix must have the same number of rows as the columns in the matrix being multiplied.

We start from the top-left corner and select the first element with a 1. We can’t select the others in the same column — they’d interfere with the sum — so we put 0s.

Then we move to the next row and next column, place a 1 there, and 0s everywhere else in that column.

We keep doing this — selecting one element per row and column — and end up with a diagonal line of 1s from top-left to bottom-right

This means the identity matrix must also have the same number of columns as the matrix being multiplied so that there are enough columns to go through all of the elements in each row

✅ Summary of Properties

An identity matrix must be:

• A square matrix with side length equal to the number of columns of the matrix being multiplied
• 1s on the main diagonal
• 0s everywhere else

🔁 Commutativity Discovery

That means multiplication with the identity matrix is commutative.

This is a nice discovery, because it wasn’t obvious at the start — we began by using it as the right matrix