Linear Algebra

Vectors, matrices, determinants and transformations

Linear Algebra brings together the core tools for working with vectors, matrices, determinants, and transformations in a clear, practical way. These formulas help students and exam candidates solve problems with confidence, and they’re just as useful in real-world work across physics, engineering, computer graphics, data science, and more. Understanding them makes it easier to describe motion, solve systems, and analyze how things change.

Determinant (2×2)

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Determinant of a 2×2 matrix.

Dot Product

\vec{a}\cdot\vec{b} = \sum_i a_i b_i = |a||b|\cos\theta

Scalar product of two vectors.

Cross Product Magnitude

|\vec{a}\times\vec{b}| = |a||b|\sin\theta

Magnitude of the vector perpendicular to two vectors.

Vector Magnitude

|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}

Length of a vector.

Eigenvalue Equation

A\vec{v} = \lambda\vec{v}

Defines eigenvalues and eigenvectors of a matrix.

Inverse of a 2×2 Matrix

A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Inverse of a 2×2 matrix (ad−bc ≠ 0).

Trace of a Matrix

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Vector Projection

\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}

Projection of vector a onto vector b.

Matrix Addition

(A+B)_{ij} = a_{ij} + b_{ij}

Matrices of equal size add entry by entry.

Scalar Multiplication

(cA)_{ij} = c\,a_{ij}

Multiply every entry of a matrix by a scalar.

Identity Matrix

AI = IA = A

The identity matrix leaves any matrix unchanged under multiplication.

Matrix Transpose

(A^T)_{ij} = a_{ji}

Reflects a matrix across its main diagonal.

Determinant (3×3)

\det = a(ei-fh) - b(di-fg) + c(dh-eg)

Determinant of a 3×3 matrix by cofactor expansion.

Matrix Trace

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Dot Product

\mathbf{a}\cdot\mathbf{b} = \sum a_i b_i

Sum of componentwise products of two vectors.

Vector Magnitude

\|\mathbf{v}\| = \sqrt{\sum v_i^2}

Length of a vector.

Angle Between Vectors

\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{\|a\|\|b\|}

Angle from the dot product and magnitudes.

Scalar Triple Product

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})

Volume of the parallelepiped spanned by three vectors.

Identity Matrix Property

AI = IA = A

Multiplying by the identity leaves a matrix unchanged.

Transpose of a Product

(AB)^T = B^T A^T

The transpose of a product reverses the order.

Rank-Nullity Theorem

\text{rank}(A) + \text{nullity}(A) = n

Relates rank and null space dimension to columns.

2×2 Matrix Product

\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}e&f\\g&h\end{bmatrix} = \begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}

Product of two 2×2 matrices.

Unit Vector

\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}

Vector of length one in the same direction.

Determinant of a Product

\det(AB) = \det(A)\det(B)

The determinant of a product is the product of determinants.