Vectors & Matrices Formulas

Matrix Addition

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

Matrix Addition combines two matrices of the same size by adding each pair of elements in matching positions; use it to combine data sets.

Matrix Multiplication

(AB)_{ij} = \sum_{k} a_{ik} b_{kj}

Matrix multiplication combines rows from the first matrix with columns of the second to produce a new matrix, useful for solving systems and transformations.

Determinant of a 2x2 Matrix

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

A single number computed from a 2x2 matrix that tells whether the matrix is invertible and helps solve systems of linear equations.

Determinant of a 3x3 Matrix

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

The determinant of a 3x3 matrix reveals whether it is invertible and how the transformation scales volume in three-dimensional space.

Inverse of a 2x2 Matrix

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

Compute the unique 2×2 matrix that reverses multiplication by another 2×2 matrix, essential for solving systems of linear equations.

Identity Matrix

AI = IA = A

The identity matrix leaves any matrix unchanged when multiplied—the matrix equivalent of 1, essential for solving linear systems and finding inverses.

Transpose of a Matrix

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

The transpose of a matrix swaps its rows and columns; use it when matrices need to match dimensions for multiplication or to solve systems of equations.

Cramer's Rule

x_i = \frac{\det A_i}{\det A}

Cramer's Rule uses determinants to solve a square system of linear equations by replacing columns of the coefficient matrix with the constants.

Gaussian Elimination

[A \mid b] \to \text{row echelon form}

Gaussian Elimination transforms a system of linear equations into row echelon form to solve for unknown variables systematically.

Rank of a Matrix

\operatorname{rank}(A) = \text{number of pivot columns}

The rank of a matrix counts its pivot columns, telling you the dimension of its column space and whether a system has a solution.

Eigenvalues and Eigenvectors

A\mathbf{v} = \lambda\mathbf{v}

Eigenvalues and eigenvectors show which vectors a matrix scales without rotating, revealing fundamental properties of linear transformations and systems.

Characteristic Polynomial

\det(A - \lambda I) = 0

Solve the characteristic polynomial equation to find a matrix's eigenvalues—the scaling factors that reveal how the matrix transforms vectors.

Vector Addition

\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, ; u_2 + v_2 \rangle

Combine two vectors by adding their components separately to find the resultant vector that represents their total effect or displacement.

Dot Product

\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta

The dot product is a scalar value that measures how much two vectors align; use it to find angles between vectors or project one vector onto another.

Cross Product

|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta

The cross product finds a vector perpendicular to two given vectors; its magnitude equals the area of the parallelogram they form.

Magnitude of a Vector

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

Magnitude of a vector is its length or size, measured as the distance from the origin; use it to find how far something has traveled overall.

Unit Vector

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

Create a vector of magnitude 1 pointing in the same direction as the original by dividing by its magnitude — use when direction matters more than distance.

Angle Between Vectors

\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}

Find the angle between two vectors using their dot product and magnitudes—essential for measuring directional relationships in physics and geometry

Vector Projection

\operatorname{proj}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2}\mathbf{v}

Vector projection breaks down one vector into a component along another's direction, essential for decomposing forces in physics and engineering.

Linear Independence

c_1\mathbf{v}_1 + \cdots + c_n\mathbf{v}_n = \mathbf{0} \implies c_i = 0

Linear Independence tells you when a set of vectors has no redundancy: none can be written as a combination of the others.

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