1
Choose and execute a solving method
Use a graph to classify and estimate, choose substitution or elimination for an exact two-variable solution, and extend elimination to three variables.
Course · Algebra
Systems and Matrices
Solve for several unknowns together, then find the notation that removes the bookkeeping. A matrix is elimination with the letters deleted, which is why this course does elimination by hand first and only then writes it down as rows.
one solution, none, or infinitely many — and nothing else
R2 → R2 − 3R1 an elimination step, written as a row operation
det A = 0 ⟺ A has no inverse
the optimum of a linear objective sits at a corner
What this course is about
- Subject
- Algebra
- Lessons
- 10
- Format
- Interactive
Overview
What you will be able to do
By the end you can choose and carry out an exact method for the two- and three-variable systems studied here, classify every outcome, and check the result against the original constraints.
2
Translate and row reduce
Write a system as an augmented matrix, perform reversible row operations, carry it to reduced row echelon form, and read unique, empty and parameterised solution sets.
3
Test and invert a coefficient matrix
Compute a determinant, use it to decide uniqueness and invertibility, construct an inverse when one exists, and verify it by multiplication.
4
Optimise over linear constraints
Build a closed, bounded feasible region, solve boundary pairs for its corners, reject infeasible crossings, and compare an objective at the surviving corners.
Lessons
10 lessons
Systems are classified and solved by hand first. That work is then turned into row operations, reduction and matrix products; determinants and inverses are added next. “Systems of Inequalities and Linear Programming” returns to two-variable systems to optimise over a bounded feasible region.
Systems of Two Linear EquationsClassify two lines as meeting once, never or everywhere, then verify a candidate in both equations.Not marked
Solving by SubstitutionChoose an efficient isolation, solve by substitution, and diagnose a contradiction or identity.Not marked
Solving by EliminationChoose the cheaper variable, eliminate it reversibly, and classify what remains.Not marked
Systems in Three VariablesPlan two eliminations of the same variable, solve the reduced system, and verify the triple.Not marked
Matrices and Row OperationsTranslate a system to an augmented matrix and apply one reversible row operation at a time.Not marked
Gaussian EliminationReduce an augmented matrix in exact fractions and classify its solution set from pivots.Not marked
Matrix ProductsDecide whether a matrix product exists, state its size, and compute each row-column entry.Not marked
Determinants and Cramer's RuleCompute a determinant, use it to test uniqueness, and apply Cramer's rule only when division is legal.Not marked
Inverse MatricesTest invertibility, construct an inverse, verify both products, and use it on the correct side.Not marked
Systems of Inequalities and Linear ProgrammingBuild a closed, bounded feasible region, test every boundary crossing, and optimise at its corners.Not marked
Recommended background
Lines, functions, and exact fraction arithmetic.
Practice suggestions
After each complete example, cover its answer and finish the faded rehearsal. The first decision is supplied; the remaining elimination, row operation, classification and check are yours.
Keep everything in fractions. Row reduction produces thirds and sevenths immediately, and a decimal here compounds into a visibly wrong answer three rows later — which is why every lab on this course is exact.
Predict the case or next operation before revealing a lab result. Then check a system in its original equations and an inverse by multiplying: a tidy derived matrix cannot certify the arithmetic that produced it.
What it does not cover
Vector spaces, linear independence, rank as a general concept, and eigenvalues. This is the matrix algebra a school course needs, not a linear algebra course.
Systems with more equations than unknowns, least squares, and numerical conditioning.
The simplex algorithm. Linear programming here is done by evaluating the objective at every corner of a feasible region you can draw.