Matrices & Linear Maps

A matrix is a linear map between vector spaces; matrix multiplication composes maps, and inverses undo them when they exist.

1. The idea

A matrix $A \in R^{m \times n}$ takes a vector $x \in R^{n}$ to $A x \in R^{m}$ , and this map is linear: $A (α x + β y) = α A x + β A y$ . Every linear map between finite-dimensional spaces can be written this way once a basis is fixed.

Matrix multiplication $A B$ represents composing the map $B$ then $A$ . This is why matrix multiplication is associative but not commutative in general: the order of transformations matters.

A square matrix $A$ is invertible when it maps $R^{n}$ bijectively onto itself; equivalently, its determinant is nonzero, its columns are linearly independent, and $A x = 0$ only for $x = 0$ .

2. Why it matters

Covariance matrices, factor loadings, transition matrices and regression design matrices are all matrices whose action as linear maps is what interviewers probe.

3. Core mathematics

Matrix-vector product

(A x)_{i} = j = 1 \sum n A_{ij} x_{j}

Transpose rule

(A B)^{T} = B^{T} A^{T}

Inverse

A^{- 1} A = A A^{- 1} = I

exists iff $det (A) \neq = 0$

Determinant of product

det (A B) = det (A) det (B)

4. How to recognize it

Think of this when…

Systems of linear equations, or 'find the transformation that maps ...'
Regression coefficients $β^= (X^{T} X)^{- 1} X^{T} y$
Any factor-model setup with loadings times factor returns

5. Worked example

Worked example

Let $A = (2111)$ . Find $A^{- 1}$ and use it to solve $A x = (3, 2)^{T}$ .

1
Compute the determinant.
$det (A) = 2 (1) - 1 (1) = 1$
2
Invert using the 2x2 formula.
$A^{- 1} = \frac{1}{1} (1 - 1 - 1 2)$
3
Apply to the target vector.
$x = A^{- 1} (3, 2)^{T} = (1 (3) - 1 (2), - 1 (3) + 2 (2)) = (1, 1)$
Answer: x = (1, 1)

6. Common mistakes

Assuming $A B = B A$ ; matrix multiplication is generally not commutative
Inverting a matrix that is singular (determinant zero) without checking first
Mixing up row and column conventions when reading dimensions

7. Quick check

$(A B)^{T}$ equals:

A square matrix is invertible iff:

8. Practice

8 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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