Mastering Matrix Basics: Operations, Transpose, and Inverses Explained

1 Matrix Basic Operations

Matrix addition follows standard properties such as commutativity, associativity, existence of a zero matrix, and additive inverses. Matrix multiplication obeys associativity, distributivity over addition, and has an identity matrix, but generally is not commutative.

Example 1 – Addition

(Illustrative example omitted.)

Example 2 – Multiplication

(Illustrative example omitted.)

Example 3 – Non‑commutativity

Matrix multiplication usually does not satisfy the commutative law; for instance, AB ≠ BA in general.

2 Matrix Transpose

The transpose of a matrix swaps its rows and columns. The transpose has several useful properties.

Properties of the Transpose

(1) (Aᵀ)ᵀ = A

(2) (A + B)ᵀ = Aᵀ + Bᵀ

(3) (kA)ᵀ = kAᵀ for any scalar k

(4) (AB)ᵀ = BᵀAᵀ

3 Identity Matrix

Properties

The identity matrix I satisfies AI = IA = A for any conformable matrix A.

I is its own transpose.

4 Elementary Matrices

An elementary matrix is obtained by applying a single elementary row operation to the identity matrix. Examples include swapping two rows or adding a multiple of one row to another. Multiplying a matrix on the left by an elementary matrix performs the corresponding row operation.

A key result is that any invertible matrix can be expressed as a product of elementary matrices.

5 Inverse Matrices

If a matrix A has a matrix B such that AB = BA = I, then B is the inverse of A, denoted A⁻¹.

Algorithm for Computing the Inverse

Form the augmented matrix [A | I] and apply a sequence of elementary row operations (i.e., left‑multiply by elementary matrices) until the left block becomes the identity matrix. The right block then becomes A⁻¹.

Example

Determine conditions under which a given matrix is non‑invertible and construct its inverse using the augmentation method.

Properties of Invertible Matrices

If A and B are invertible, then (AB)⁻¹ = B⁻¹A⁻¹.

If A is invertible, then (Aᵀ)⁻¹ = (A⁻¹)ᵀ.

If A and B are symmetric and invertible and AB is symmetric, then (AB) is also symmetric.