Matrix Transformations and Linear Algebra

Learning objectives

• Describe matrices by their size and carry out addition, subtraction, scalar multiplication, multiplication and transposition.
• Decide when a matrix operation is defined, and explain why AB ≠ BA in general.
• Interpret a matrix as a linear transformation and use homogeneous coordinates for translations.
• Interpret the determinant as an area (or volume) scale factor with a sign for orientation.
• Find the inverse of a 2×2 matrix and explain when no inverse exists.
• Explain eigenvalues and eigenvectors with A v = λ v, including matrices with no real eigenvectors.

What a matrix is

A matrix is a rectangular array of numbers. Its size (or dimensions) is written rows × columns: a 2×3 matrix has 2 rows and 3 columns. The entry in row i and column j is written a_ij. Asquare matrix has as many rows as columns, and the identity matrix I has 1s on its diagonal and 0s elsewhere: multiplying by I changes nothing.

Matrix operations

• Addition and subtraction: only for matrices of the same size; add or subtract matching entries.
• Scalar multiplication: k·A multiplies every entry by the number k.
• Transpose: Aᵀ swaps rows and columns, so (Aᵀ)_ij = a_ji and an m×n matrix becomes n×m.

Matrix multiplication

The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m×n and B is n×p, then AB is m×p, and each entry is a row of A “dotted” with a column of B:

(AB)_ij = a_i1b_1j + a_i2b_2j + … + a_inb_nj

Matrix multiplication is associative ((AB)C = A(BC)) but not commutative: AB and BA are usually different, and one can exist while the other does not.

Matrices as transformations

Multiplying a column vector by a matrix transforms it. In this lab, points are column vectors and the matrix acts on the left: p′ = M p. The columns of a 2×2 matrix show where the basis vectors î = (1, 0) and ĵ = (0, 1) land, and every other point follows: (x, y) ↦ x·î′ + y·ĵ′. Such a linear transformation keeps the origin fixed and turns straight lines into straight lines and parallel lines into parallel lines.

To apply T₁ and then T₂, multiply the matrices as T₂ · T₁: the transformation applied first sits next to the vector. Because multiplication is not commutative, the order changes the result. (Some books and graphics tools use row vectors, p′ = p M, which reverses the order of every product; this lab uses column vectors throughout.)

Homogeneous coordinates

A translation moves the origin, so it is not a linear transformation and no 2×2 matrix can do it. The trick is to write each point as a 3-component column (x, y, 1). Then a translation becomes the 3×3 matrix [1 0 tₓ; 0 1 t_y; 0 0 1], and rotations, scalings, reflections and shears put their 2×2 matrix in the top-left corner. Every combination of these is one 3×3 matrix product, which is how computer graphics handles them.

The determinant

For a 2×2 matrix, det [a b; c d] = ad − bc. Geometrically, |det A| is the factor by which A scales areas: the unit square becomes a parallelogram of area |det A|. The sign shows orientation: a negative determinant means the transformation includes a reflection, so shapes come out as mirror images. If det A = 0 the plane is squashed onto a line or a point. For 3×3 matrices the determinant scales volumes; it can be calculated by expanding along a row: det A = a₁₁·det M₁₁ − a₁₂·det M₁₂ + a₁₃·det M₁₃, where M_1j is A with row 1 and column j removed.

The inverse

The inverse A⁻¹ undoes A: A·A⁻¹ = A⁻¹·A = I. Only square matrices with det A ≠ 0 have one (they are called invertible or non-singular). For a 2×2 matrix,A⁻¹ = 1/(ad − bc) · [d −b; −c a]. In general A⁻¹ = adj(A)/det A, where the adjugate adj(A) is the transpose of the matrix of cofactors. A singular matrix (det A = 0) has no inverse because it loses information: many points land on the same place, so you cannot tell where they came from.

Eigenvalues and eigenvectors

An eigenvector of A is a non-zero vector v that A only stretches, shrinks or reverses, without turning it off its line: A v = λ v. The number λ is its eigenvalue. For a 2×2 matrix the eigenvalues solve λ² − (a + d)λ + (ad − bc) = 0, the characteristic equation.

• Two different real roots: two eigen-directions (for example a stretch, or the symmetric matrix [2 1; 1 2]).
• A repeated root: either every vector is an eigenvector (A = λI) or there is only one eigen-direction (a shear).
• Complex roots: no real eigenvectors at all. A rotation by 90° turns every vector, so nothing keeps its direction.

The product of the eigenvalues equals det A, and their sum equals the trace a + d.

3D transformations

In 3D, 3×3 matrices scale, rotate and reflect objects about the origin. The lab builds M = F·R·S: scale first, then rotate about one axis, then reflect in a coordinate plane. Its determinant scales volume, and a negative determinant produces a mirror image that no rotation could reach. Orbiting the camera changes your viewpoint only: the object and its matrix stay the same.

Formula reference

Real-world applications

• Computer graphics and games: every frame, 4×4 homogeneous matrices position, rotate and project millions of points onto the screen.
• Robotics: chains of transformation matrices locate a robot arm’s hand from its joint angles.
• Engineering: large systems of linear equations for structures and circuits are solved with matrices.
• Data science: principal component analysis finds the eigenvectors of a data set’s covariance matrix to reveal its main directions of variation.
• Search and networks: Google’s original PageRank was an eigenvector of a huge matrix of links between pages.
• Physics: eigenvalues give the energy levels in quantum mechanics and the natural vibration frequencies of bridges and buildings.

Interesting facts

• The word “matrix” was introduced by James Joseph Sylvester in 1850; Arthur Cayley defined matrix multiplication in 1858.
• Chinese mathematicians solved systems of equations with grids of numbers, much like matrices, about 2,000 years ago in The Nine Chapters on the Mathematical Art.
• Multiplying two n×n matrices the schoolbook way takes about n³ multiplications; cleverer algorithms, starting with Strassen’s in 1969, need fewer.
• Graphics cards are, at heart, machines for multiplying matrices very quickly, which is why they also power modern AI.

Common misconceptions

• “AB = BA.” Usually false, and one product may not even be defined.
• “Multiply matrices entry by entry.” The matrix product combines rows with columns.
• “Any square matrix has an inverse.” Only if its determinant is not zero.
• “A negative determinant means negative area.” Area scales by |det|; the sign only records a flip.
• “Every matrix has real eigenvectors.” Rotations, for example, have none.
• “A matrix transformation is the same as adding vectors.” Adding a vector only shifts points; a matrix can rotate, stretch, shear and reflect them.

Important terms

Matrix

A rectangular array of numbers with a size rows × columns.

Identity matrix

The square matrix I with 1s on the diagonal; AI = IA = A.

Transpose

The matrix with rows and columns swapped.

Determinant

A number from a square matrix: its signed area or volume scale factor.

Inverse

The matrix that undoes A; exists only when det A ≠ 0.

Linear transformation

A map that keeps the origin fixed and preserves addition and scaling of vectors.

Homogeneous coordinates

Writing (x, y) as (x, y, 1) so translations become matrix products.

Eigenvector and eigenvalue

A non-zero v and number λ with A v = λ v.

Key takeaways

• Check sizes first: AB needs columns of A = rows of B.
• A matrix is a transformation; its columns show where the basis vectors go.
• Order matters: “T₁ then T₂” is T₂·T₁ with column vectors.
• |det| scales area or volume; a negative sign flips orientation; zero means no inverse.
• Eigenvectors keep their line; eigenvalues tell how much they stretch, and some matrices have none that are real.