How does the Gauss method determine whether a linear system has one, none, or infinitely many solutions?

Row-reduce the augmented matrix: a pivot in every variable gives one solution, a contradictory row gives none, and free variables give infinitely many solutions.

What does a negative discriminant mean in a quadratic equation?

For real coefficients, a negative discriminant means the quadratic has no real roots; it has two complex conjugate roots.

When can Cramer's rule be used to solve a system of equations?

It can be used for a square system when the determinant of the coefficient matrix is nonzero, ensuring a unique solution.

Free Course Image Math Fundamentals Algebra

Free online course Math Fundamentals Algebra

Duration of the online course: 2 hours and 9 minutes

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Strengthen algebra skills with a free online course: equations, matrices, determinants and Gauss method, plus exercises to practice and improve fast.

In this free course, learn about

Use matrices to model and solve linear systems (Ax=b) in linear algebra
Solve polynomial equations; apply Ruffini’s rule for synthetic division
Interpret quadratic discriminant; b^2−4ac<0 implies no real roots (complex pair)
Transform linear equations via equivalent operations and row operations
Classify linear systems: determined, indeterminate, or inconsistent
Apply Gauss elimination, including systems with parameters
Define matrices; recognize symmetric matrices (a_ij = a_ji)
Perform matrix operations and know key properties and common pitfalls
Identify regular/invertible matrices and their implications
Solve matrix equations (e.g., B·X=D) and determine compatible matrix sizes
Compute determinants (2x2 and beyond) and use determinant properties
Understand matrix rank (range) and compute/estimate maximum rank
Compute inverses using adjugate/cofactors; find specific adjoints
Apply Cramer’s rule; det(A)≠0 implies A is invertible and solution is unique

About the free online course

Build a solid algebra foundation and gain confidence solving problems step by step in this free online course focused on the essentials you need for school, exams, and future STEM studies. Whether you feel stuck with equations or want to understand what is really happening behind common procedures, this training helps you move from memorizing rules to reasoning clearly and checking your results with certainty.

You will start by sharpening your handling of algebraic equations, including quadratic equations and the meaning of the discriminant when analyzing the number and type of solutions. From there, you will develop a clear approach to linear equations, learning how to transform them into equivalent forms without changing their solution set. This ability becomes crucial when you move from single equations to systems, where identifying the nature of solutions matters as much as calculating them.

Next, you will work with structured techniques for solving linear systems, especially the Gauss method, including situations with parameters where solutions can change depending on given values. This is a practical skill that supports problem solving across science, economics, and technology. You will then connect these ideas to matrices: what they are, how to operate with them correctly, what makes a matrix regular or invertible, and how matrix equations are formulated and solved in a consistent way.

As you progress, determinants and rank provide the tools to understand when a system has a unique solution, infinitely many, or none, and why. You will also explore inverse matrices, adjoints, and Cramer’s rule, gaining an intuitive view of what conditions guarantee solvability in matrix form. Throughout the course, the included exercises reinforce learning by prompting you to interpret results, avoid common mistakes, and strengthen mathematical maturity.

Course content

Video class: Presentation | 1/14 | UPV 03m
Exercise: In the context of the course on Linear Algebra, how can matrices be applied to solve systems of linear equations?

Video class: Algebraic equations with one unknown. Ruffini's rule | 2/14 | UPV 13m
Exercise: What can be concluded if the discriminant (b^2 - 4ac) of a quadratic equation is negative?

Video class: Linear equations 2 | 3/14 | UPV 10m
Exercise: What kind of transformations can be used to obtain equivalent linear equations?
Video class: System of linear equations 2 | 4/14 | UPV 09m
Exercise: Given a system of linear equations: 3x + 4y = 10 and 6x + 8y = 20, determine the type of system in terms of solutions.
Video class: Gauss method | 5/14 | UPV 12m
Exercise: Which of the following best describes a consistent determined system when using the Gauss method to solve linear equations?
Video class: Examples of systems of linear equations with parameters using the Gauss method | 6/14 | UPV 09m
Exercise: Which of the following describes a consistent indeterminate system when using the Gauss method to solve a linear equation system with parameters?

Video class: Definition of Matrix | 7/14 | UPV 07m
Exercise: Consider a 3x3 matrix with elements a_ij. If the matrix is symmetric, which condition must it satisfy?
Video class: Operations with matrices | 8/14 | UPV 08m
Exercise: Which of the following statements about matrix operations is incorrect?

Video class: Regular matrices | 9/14 | UPV 11m
Exercise: Which of the following statements is true regarding regular or invertible matrices?
Video class: Matrix equations | 10/14 | UPV 07m
Exercise: Given the matrix equation ‘B * X = D’ where B is a 3x3 matrix and D is a 3x3 matrix, what is the size of the matrix X that ensures successful computation of this matrix equation?

Video class: Determinants | 11/14 | UPV 13m
Exercise: If a 2x2 square matrix is given as \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), how is its determinant calculated?
Video class: Matrix Range | 12/14 | UPV 05m
Exercise: Given a matrix C with dimensions 4x5, what is the maximum possible rank of this matrix?
Video class: Inverse matrix calculation | 13/14 | UPV 06m
Exercise: What is the adjoint of the element a_23 in a 3x3 matrix if the complementary submatrix determinant is -5?

Video class: Cramer's rule | 14/14 | UPV 10m
Exercise: Consider a 3x3 linear system of equations which can be written in matrix form as A * X = B, where A is the coefficient matrix, X is the vector of unknowns (x, y, z), and B is the vector of independent terms. If the determinant of A is not zero, what does this imply about matrix A?

Free online courses on Intermediate Algebra

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Explore free online Intermediate Algebra courses designed to strengthen your skills in equations, inequalities, functions, polynomials, rational expressions, radicals, and graphing. Learn at your own pace with practical lessons and earn a certificate upon completion. Build the algebra knowledge needed for college, exams, and career growth today.