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Linear Algebra Matrices and Determinants cheat sheet - grade college

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Math Grade college

Linear Algebra Matrices and Determinants Cheat Sheet

A printable reference covering matrix operations, inverses, determinants, rank, nullity, eigenvalues, LU, QR, and SVD factorizations for college.

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This cheat sheet covers the essential matrix tools used in college linear algebra, including matrix arithmetic, inverses, determinants, rank, nullity, and common factorizations. Students need these ideas to solve systems of linear equations, analyze linear transformations, and work with data, geometry, and differential equations. A formula-forward reference helps connect computational procedures with the structure behind them.

Key Facts

  • Matrix addition and scalar multiplication are entrywise: if A=[aij]A = [a_{ij}] and B=[bij]B = [b_{ij}], then A+B=[aij+bij]A+B = [a_{ij}+b_{ij}] and cA=[caij]cA = [ca_{ij}].
  • Matrix multiplication is defined by (AB)ij=∑k=1naikbkj(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} when AA is m×nm \times n and BB is n×pn \times p.
  • The inverse of a nonsingular square matrix satisfies AA−1=A−1A=IAA^{-1} = A^{-1}A = I, and for 2×22 \times 2 matrices, [abcd]−1=1ad−bc[d−b−ca]\begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix} when ad−bc≠0ad-bc \ne 0.
  • For a 2×22 \times 2 matrix, det⁡[abcd]=ad−bc\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc.
  • A square matrix AA is invertible exactly when det⁡(A)≠0\det(A) \ne 0, rank⁡(A)=n\operatorname{rank}(A)=n, and the equation Ax=0Ax=0 has only the trivial solution.
  • The rank-nullity theorem states that for an m×nm \times n matrix AA, rank⁡(A)+nullity⁡(A)=n\operatorname{rank}(A)+\operatorname{nullity}(A)=n.
  • Eigenvalues satisfy the characteristic equation det⁡(A−λI)=0\det(A-\lambda I)=0, and eigenvectors satisfy Av=λvAv=\lambda v with v≠0v \ne 0.
  • Common factorizations include A=LUA=LU for elimination, A=QRA=QR for orthogonal decomposition, and A=UΣVTA=U\Sigma V^{T} for singular value decomposition.

Vocabulary

Matrix
A rectangular array of numbers or symbols arranged in rows and columns, often used to represent a linear transformation or system of equations.
Determinant
A scalar value det⁡(A)\det(A) assigned to a square matrix that measures signed volume scaling and indicates whether the matrix is invertible.
Rank
The rank of a matrix is the dimension of its column space, equal to the number of pivot columns in its row echelon form.
Nullity
The nullity of a matrix is the dimension of the solution space of Ax=0Ax=0, equal to the number of free variables.
Eigenvalue
An eigenvalue λ\lambda is a scalar for which there exists a nonzero vector vv satisfying Av=λvAv=\lambda v.
Factorization
A matrix factorization rewrites a matrix as a product of simpler matrices, such as A=LUA=LU, A=QRA=QR, or A=UΣVTA=U\Sigma V^{T}.

Common Mistakes to Avoid

  • Multiplying matrices entry by entry, which is wrong because matrix multiplication uses row-column dot products: (AB)ij=∑k=1naikbkj(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}.
  • Assuming AB=BAAB=BA, which is wrong because matrix multiplication is generally not commutative and the products may differ or one product may be undefined.
  • Using the inverse formula when det⁡(A)=0\det(A)=0, which is wrong because a square matrix is invertible only if det⁡(A)≠0\det(A)\ne 0.
  • Confusing rank with the number of rows, which is wrong because rank counts independent columns or pivots and can be less than both the row count and column count.
  • Treating every matrix as diagonalizable, which is wrong because diagonalization requires enough linearly independent eigenvectors to form a basis.

Practice Questions

  1. 1 Compute ABAB for A=[1234]A=\begin{bmatrix}1&2\\3&4\end{bmatrix} and B=[2015]B=\begin{bmatrix}2&0\\1&5\end{bmatrix}.
  2. 2 Find det⁡(A)\det(A) and decide whether AA is invertible for A=[4726]A=\begin{bmatrix}4&7\\2&6\end{bmatrix}.
  3. 3 If an m×5m \times 5 matrix has rank⁡(A)=3\operatorname{rank}(A)=3, find nullity⁡(A)\operatorname{nullity}(A) using rank⁡(A)+nullity⁡(A)=n\operatorname{rank}(A)+\operatorname{nullity}(A)=n.
  4. 4 Explain why a matrix with two identical rows must have determinant 00 and cannot be invertible.

Understanding Linear Algebra Matrices and Determinants

A useful way to understand a matrix is as a machine that changes vectors. A vector can represent a position, a velocity, a list of prices, or measurements from an experiment. The matrix tells each output component how to combine input components.

In geometry, a matrix may stretch a shape, rotate it, reflect it, or shear it into a slanted form. The columns show where the basic direction vectors end up. This column view makes multiplication less mysterious.

When one matrix follows another, the rightmost transformation happens first. For that reason, changing the order usually changes the result. This matters when a graphics program rotates an object before moving it, rather than moving it before rotation.

A determinant gives geometric information about a square transformation. Its absolute value tells how much area in two dimensions, or volume in three dimensions, is scaled. A determinant with absolute value two doubles area.

A determinant of zero means that some dimension has been flattened. For example, a two dimensional region might be sent onto a line. Once this happens, different input vectors can produce the same output.

The original input cannot be recovered uniquely. The sign of a determinant records orientation. A negative value means the transformation has flipped the shape, much like a reflection in a mirror.

Rank measures how many independent directions remain after a matrix acts. It is not simply the number of nonzero rows or columns before calculation. Row reduction reveals the number of pivot positions, which gives the rank.

Nullity describes how many independent input directions are sent to the zero vector. These ideas explain whether a system has one solution, no solution, or many solutions. They are important in data analysis because real data often contain repeated or dependent information.

A table may have many columns, yet only a few may contribute genuinely new information. Students should watch for rounding errors here. A calculator may display a very small number instead of zero, which can make rank decisions less clear.

Eigenvectors are special directions that do not turn away from their original line under a transformation. They may grow, shrink, or reverse direction. The matching eigenvalue gives that scale factor.

This is useful for studying repeated changes, such as population models, vibration patterns, web page ranking, and systems that change over time. Factorizations provide practical ways to organize difficult calculations. Lower upper factorization records the row elimination process and makes repeated system solving faster.

Orthogonal factorization uses perpendicular directions, which are stable in numerical work. Singular value decomposition identifies the strongest directions in any data table, even when the table is not square.

When learning these methods, focus on what each factor does. Memorizing a procedure without tracking dimensions, pivot choices, or geometric meaning leads to common mistakes.