## 17Calculus Precalculus - Identity Matrices

##### 17Calculus

Before we get started with matrix inverses, we need to understand what an identity matrix is. The identity matrix is a square matrix with zeroes everywhere EXCEPT on the diagonal. On the diagonal we have all ones. So a 2x2 identity matrix is
$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$.
A 3x3 identity matrix is
$$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$.

Identity matrices work like the number $$1$$ for numbers, i.e. $$1 \cdot 3 = 3 \cdot 1 = 3$$. So any number you multiply by 1 is the number itself. The same holds for identity matrices. So if we have a 2x2 matrix A, $$A\cdot I_2 = I_2 \cdot A = A$$. For a 3x3 matrix $$B\cdot I_3 = I_3 \cdot B = B$$.

Identity Matrices Summary

1. Identity matrices are always square, i.e. they have the same number rows as columns.

2. Identity matrices always have ones in the diagonal and zeroes everywhere else.

3. Although matrix multiplication, in general, is not commutative, when multiplying a matrix by the identity matrix, it can be multiplied on either side (assuming the other matrix is square also).

4. As you may have picked up above, books and instructors will often write a subscript on the capital I indicating the size of the identity matrix. So a matrix noted as I2 is a 2x2 identity matrix.

We will hold off working any practice problems until you get the section on Matrix Inverses where you will work with identity matrices.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

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