MAT 242 Elementary Linear Algebra
The Complete Exam Bible · Fall 2026MAT 242 Overview
- Arizona State University
- Fall 2026, Session C
- Mathematics
- Lower-division undergraduate
- Three in-person tests
MAT 242 at Arizona State University introduces matrices, systems of linear equations, determinants, vector spaces, linear transformations and eigenvalues, with the emphasis on computational skill.
- Assessed by Test 1, Test 2 and the Final Exam at 25% each, plus written and Edfinity homework.
- Hardest step Choosing the right basis rule for null, row and column spaces, and the change-of-basis order.
- How to prepare Redo each worked example by hand with labelled row operations, then check by substitution.
- No calculator on Test 1 Practise elimination, inverses and determinants by hand before October 1.
How MAT 242 is assessed
| Component | Weight | Format |
|---|---|---|
| Test 1 | 25% | In person, 50 minutes, no calculator; sections 1.1 to 3.1 |
| Test 2 | 25% | In person, 50 minutes, non-CAS graphing calculator; sections 4.2 to 5.2 |
| Final Exam (Test 3) | 25% | In person, 1 hour 50 minutes, not comprehensive |
| Written homework | 15% | Weekly written sets submitted as one PDF |
| Edfinity homework | 10% | Online homework, one assignment per section |
Weights are from the Fall 2026 syllabus and total 100%. For Tests 1 and 2 the syllabus adds a 5% bonus to the test score when the graded test is collected on the first or second hand-back day.
Assessment structure
Segment widths reproduce the published weights: Test 1, Test 2, the Final Exam, written homework and Edfinity homework, totalling 100%.
Current MAT 242 dates
| Date | Item | Control |
|---|---|---|
| 1 October 2026 | Test 1 | Fall 2026 Canvas test schedule. |
| 4 November 2026 | Course withdrawal deadline | Fall 2026 Canvas syllabus page. |
| 12 November 2026 | Test 2 | Fall 2026 Canvas test schedule. |
| 10 December 2026 | Final Exam (Test 3) | Fall 2026 syllabus schedule; the time depends on your section. |
Dates are taken from the Fall 2026 syllabus and Canvas course pages. Confirm exact deadlines and submission settings in the live LMS.
What MAT 242 covers
Fourteen chapters follow the course notes from linear systems to orthogonality, grouped by the test that assesses them.
Linear Systems and Gaussian Elimination
Elimination, back substitution, augmented matrices and the three row operationsWhy the course starts with systems
Almost every later calculation in the course, from inverses to eigenvectors to least squares, ends with row reducing a matrix. This chapter builds that habit from the ground up.
- Elimination and back substitution
- Consistent, inconsistent and equivalent systems
- Augmented matrices
- The three row operations
- Row echelon form
Gauss-Jordan Elimination and Parameterization
Reduced row echelon form, pivots, free variables and describing infinitely many solutionsOne more pass: reduced row echelon form
Gauss-Jordan elimination continues where Gaussian elimination stops. After the staircase is built, the entries above each leading 1 are cleared too, which leaves reduced row echelon form.
- Reduced row echelon form
- Pivots, lead variables and free variables
- Counting solutions from an echelon form
- Parameterizing infinitely many solutions
- Systems with an unknown coefficient
Matrix Arithmetic and Matrix Multiplication
Entries and dimensions, sums and scalar multiples, the row-by-column product, and AX = BMatrices become objects
This chapter treats a matrix as a thing you can compute with. An m × n matrix has m rows and n columns; its entry in row i and column j is written with two subscripts.
- Entries, rows, columns and dimensions
- Matrix equality, addition and scalar multiples
- When a product is defined
- Computing AB entry by entry
- Writing a system as AX = B
Inverses of Matrices
The 2 × 2 formula, the [A | I] procedure, solving X = A⁻¹B, and the rules for inversesWhat an inverse is
The inverse of a square matrix A is the matrix that undoes it: multiplying in either order returns the identity. When it exists it is written A⁻¹, and A is called invertible or nonsingular; otherwise A is singular.
- Invertible and singular matrices
- The 2 × 2 inverse formula
- Inverting with [A | I]
- Solving AX = B with the inverse
- Inverse of a product and cancellation rules
Determinants by Expansion by Minors
Minors, cofactors and the checkerboard, Sarrus's method, properties, and solving det A = 0A number that decides invertibility
For a 2 × 2 matrix, ad − bc tells you whether an inverse exists. The determinant generalises that number to every square matrix: det A is nonzero exactly when A is invertible.
- Minors and cofactors
- The checkerboard of signs
- Expanding along the sparsest row or column
- Sarrus's method for 3 × 3 matrices
- Properties of determinants
Subspaces, Spanning Sets and Linear Independence
Vector spaces and the Subspace Test, span and membership, and linear dependenceFrom answers to structure
The course notes describe this part of the textbook as its most abstract: the focus moves from computing a single answer to describing a whole set. Rⁿ, the set of column vectors with n entries, is the model.
- Vector spaces and subspaces
- The Subspace Test
- Linear combinations and span
- Deciding membership in a span
- Linear dependence and independence
Bases, Dimension and Coordinates
Bases and dimension from a spanning set, coordinates for an ordered basis, and change of basisA basis does two jobs
A basis for a subspace spans it and is independent, so it generates every vector with nothing wasted. Subspaces have many bases, but they all contain the same number of vectors, and that number is the dimension.
- Basis and dimension
- Subspaces of the plane by dimension
- Finding a basis from a spanning set
- Coordinate vectors for an ordered basis
- The change-of-basis matrix
Row Space, Column Space and Null Space
Null space and nullity, row and column space bases, rank, and rank plus nullityThree subspaces attached to every matrix
Each matrix A comes with three subspaces.
- Null space and nullity
- Row space from the reduced rows
- Column space from the original pivot columns
- Rank and full rank
- Rank plus nullity equals the number of columns
Eigenvalues and Eigenvectors
A coupled-springs motivation, the characteristic polynomial, eigenspaces, and quick checksWhere the problem comes from
The course introduces eigenvalues through two masses hanging on springs.
- The coupled-springs motivation
- Eigenvalues from det(A − λI) = 0
- Eigenspaces as null spaces
- Triangular matrices and repeated eigenvalues
- Trace and determinant checks
Diagonalization, Matrix Powers and Jordan Form
When A = PDP⁻¹ exists, building P and D, powers of a matrix, and optional Jordan chainsThe diagonalization test
An eigenvalue's multiplicity is its exponent in the factored characteristic polynomial, and its eigenspace dimension can be smaller but never larger.
- Multiplicity versus eigenspace dimension
- Building P and D
- Matrix powers through PDⁿP⁻¹
- Explaining a non-diagonalizable matrix
- Generalized eigenvectors and Jordan blocks
Orthogonal Vectors and the Gram-Schmidt Algorithm
Dot products and orthogonality, orthogonal complements, orthogonal coordinates, Gram-Schmidt, normalisingPerpendicular in any dimension
The dot product measures angles, and the course needs only right angles: two vectors are orthogonal when their dot product is zero.
- Dot product, length and orthogonality
- The orthogonal complement as a null space
- Coordinates in an orthogonal basis
- The Gram-Schmidt algorithm
- Orthonormal bases
Orthogonal Projections and Least Squares
The normal equation, projection and closest vector, and least squares solutionsProjection without an orthogonal basis
When W is described by an ordinary basis, put the basis vectors into the columns of A.
- Projection with an ordinary basis
- The normal equation
- The closest vector in a subspace
- Least squares solutions of inconsistent systems
- Checking a least squares answer
Curve Fitting with Linear Algebra
Exact fits through data points, best-fitting lines and parabolas, and log and circle tricksUnknown coefficients enter linearly
Curve fitting works because the unknowns are the coefficients, not x. In y = ax² + bx + c the unknowns a, b and c appear linearly, so each data point supplies one linear equation.
- Exact curve fitting through data points
- Models that are linear in their coefficients
- Least squares lines and linear regression
- Restricted parabolas and log transforms
- Fitting a circle
Linear Transformations and Their Matrices
Linearity, the matrix of a transformation from images of a basis, kernel and imageFunctions that respect linear combinations
A linear transformation sends sums to sums and scalar multiples to scalar multiples.
- The definition of a linear transformation
- Columns as images of basis vectors
- A = NM⁻¹ from images of any basis
- Kernel as a null space
- Image as a column space
This guide follows the course notes section by section and reworks every method with new numbers, organised around the three equally weighted tests.
Block one: systems and matrices
The first five chapters cover everything on Test 1: Gaussian and Gauss-Jordan elimination, counting and parameterizing solutions, matrix arithmetic and products, inverses by the [A | I] method, and determinants by expansion by minors.
Test 1 is taken without a calculator, so the worked examples keep the arithmetic manageable by hand and show every labelled row operation.
Block two: vector spaces and eigenvalues
Chapters 6 to 10 cover Test 2: span and linear independence, bases, dimension and coordinates, change of basis, the null, row and column spaces with rank and nullity, eigenvalues and eigenvectors, and diagonalization with matrix powers.
A non-CAS graphing calculator is allowed from here on, and the guide shows what must still be written down.
Block three: orthogonality and applications
Chapters 11 to 13 cover the Final Exam topics: orthogonal complements, Gram-Schmidt and orthonormal bases, projections, least squares and curve fitting.
Chapter 14 covers linear transformations, whose place on the Final differs between course pages, so confirm it on Canvas.
Worked example · free
Decide invertibility, then invert
- 2det = 1·k − 2·3 = k − 6, so the matrix is invertible exactly when k ≠ 6.
- 2For k = 7 the determinant is 1; swap the diagonal and negate the other entries.
- 1Check: [1 2; 3 7] times [7 −2; −3 1] gives the identity.
Key terms
- Augmented Matrix
- An augmented matrix stores a linear system's coefficients and, after a bar, its right-hand sides.
- Reduced Row Echelon Form
- Reduced row echelon form is the unique fully reduced matrix with a leading 1 in each pivot column and zeros elsewhere in those columns.
- Pivot Column
- A pivot column contains a leading 1 of the reduced matrix and marks a lead variable.
- Inverse Matrix
- An inverse matrix undoes A, so that multiplying by it in either order gives the identity.
- Determinant
- A determinant is a number computed from a square matrix that is nonzero exactly when the matrix is invertible.
- Span
- A span collects every weighted sum of a given list of vectors, and it is always a subspace.
- Linear Independence
- Linear independence means the only combination of the vectors equal to zero uses all-zero weights.
- Coordinate Vector
- A coordinate vector lists the unique weights that build a vector from an ordered basis.
- Eigenvector
- An eigenvector is a nonzero vector that a matrix only stretches, so Av equals a multiple of v.
- Orthogonal Basis
- An orthogonal basis is a basis whose vectors are pairwise perpendicular, which makes coordinates quick to compute.
- Orthogonal Projection
- An orthogonal projection is the vector in a subspace closest to a given vector, with a perpendicular error.
- Change of Basis
- A change of basis converts the coordinates of the same vector from one ordered basis to another.
MAT 242 FAQ
How is the grade made up?
Three tests count 25% each, the third being the Final Exam, written homework counts 15% and Edfinity online homework 10%. Letter grades use a 10-point scale with a possible curve.
Can I use a calculator on the tests?
Not on Test 1, which allows no calculator of any kind. Test 2 and the Final allow a graphing calculator without symbolic algebra, so TI-89, TI-92, nSpire CAS and keyboard models are excluded.
Is the Final Exam cumulative?
The syllabus states that the Final Exam is not comprehensive, which is also why no test score is dropped. It concentrates on orthogonality, projections, least squares and curve fitting.
Which topic causes the most trouble?
Vector-space ideas in the second block, especially choosing the right basis rule for row, column and null spaces and the direction of a change-of-basis matrix. Both are drilled with checks in this guide.
How precise do test answers need to be?
Give exact values, fractions included, or decimals with at least four digits after the point unless a question says otherwise. Round only once, at the end of the calculation.
What counts as showing work on a test?
Every row operation labelled, the matrices you enter into a calculator written down with the results, and a boxed final answer. Test instructions warn that work shown in insufficient detail can earn no credit.
Are linear transformations on the Final Exam?
Course pages disagree: the Canvas test table and one sample final include them, while the Final Exam skills list does not. Chapter 14 prepares you either way; confirm with your instructor.
Questions students ask in each MAT 242 chapter
- Why can a row be multiplied by any number except zero?
- Multiplying an equation by zero erases its information and can create new solutions. Any nonzero multiple keeps exactly the same solution set, which is the property every legal row operation must preserve. Chapter 01 · Linear Systems and Gaussian Elimination
- Does a row of zeros mean there are infinitely many solutions?
- Not by itself. An all-zero row means one equation was redundant. Infinitely many solutions requires a coefficient column without a pivot, so finish the reduction and count free variables before deciding. Chapter 02 · Gauss-Jordan Elimination and Parameterization
- How can I tell quickly whether two matrices can be multiplied?
- Write their sizes side by side. The inner numbers must agree, the columns of the left factor against the rows of the right one, and the outer numbers give the size of the product. Chapter 03 · Matrix Arithmetic and Matrix Multiplication
- What is the inverse of a product of two matrices?
- It is the product of the two inverses in the opposite order, B⁻¹A⁻¹ for AB. Writing A⁻¹B⁻¹ is a common slip that is only right when the matrices commute. Chapter 04 · Inverses of Matrices
- Can Sarrus's method be used on a 4 × 4 matrix?
- No. The diagonal shortcut is only valid for three-by-three matrices. For a four-by-four, expand by minors along a convenient row or column and then use Sarrus on the three-by-three minors if you like. Chapter 05 · Determinants by Expansion by Minors
- Is the set of solutions of Ax = b a subspace?
- Only when b is the zero vector. With a nonzero right-hand side the zero vector is not a solution, so the set fails the Subspace Test, although it is still a shifted copy of the null space. Chapter 06 · Subspaces, Spanning Sets and Linear Independence
- Why does the order of basis vectors matter for coordinates?
- Coordinates are listed in the same order as the basis vectors. Reordering the basis permutes the coordinate entries, so the course writes ordered bases with parentheses rather than set braces. Chapter 07 · Bases, Dimension and Coordinates
- Why take column-space vectors from the original matrix instead of the RREF?
- Row operations change the column space, so the RREF's columns usually span a different subspace. The RREF is only used to locate the pivot positions; the basis vectors come from A itself. Chapter 08 · Row Space, Column Space and Null Space
- Can zero be an eigenvalue?
- Yes. Zero is an eigenvalue exactly when A itself is singular, because then A sends some nonzero vector to zero. Eigenvectors, on the other hand, are never allowed to be the zero vector. Chapter 09 · Eigenvalues and Eigenvectors
- Does the order of eigenvalues in D matter?
- Any order works, provided each eigenvalue sits in the same position as its eigenvector's column in P. Reordering one without the other gives a wrong factorization. Chapter 10 · Diagonalization, Matrix Powers and Jordan Form
- When can coordinates be found with dot products instead of an inverse?
- Only when the basis is orthogonal. Each coordinate is then the dot product of the basis vector with u divided by that basis vector's dot product with itself; on a non-orthogonal basis the formula gives the wrong answer. Chapter 11 · Orthogonal Vectors and the Gram-Schmidt Algorithm
- Is the least squares solution the same as the projection?
- No. The least squares solution holds the weights, one per variable. Multiplying it by A gives the projection of b onto the column space, a different vector with one entry per equation. Chapter 12 · Orthogonal Projections and Least Squares
- Why can a parabola be fitted with linear algebra when it is not a straight line?
- The unknowns a, b and c multiply known quantities x², x and 1, so each point gives an equation that is linear in the unknowns, even though the curve itself bends. Chapter 13 · Curve Fitting with Linear Algebra
- How do I recognise that a function is not linear?
- Check where it sends the zero vector and whether it respects sums. A shift such as adding a fixed nonzero vector sends zero somewhere else, so it cannot be linear. Chapter 14 · Linear Transformations and Their Matrices
How to study for the exam
Work in three blocks, one per test. For each chapter, read the worked example, redo it with the numbers covered, then do the related Edfinity set and the written homework in the course's written style: labelled operations, boxed answers, a check at the end.
Before each test, sit one sample test under the real time limit, 50 minutes for Tests 1 and 2 and 1 hour 50 minutes for the Final, and rework every item that lost points within a day.
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