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HomeHomework HelpmathematicsDeterminants and Cramer’s Rule

Determinants and Cramer’s Rule

Determinants are scalar values that provide important information about a matrix, such as whether it is invertible, while Cramer’s Rule is a mathematical theorem that provides an explicit solution to a system of linear equations using determinants.

intermediate
3 hours
Mathematics
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Overview

Determinants and Cramer's Rule are fundamental concepts in linear algebra that help solve systems of linear equations. Determinants provide a way to assess the properties of matrices, such as whether they are invertible, while Cramer's Rule offers a method to find solutions to these systems using de...

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Key Terms

Matrix
A rectangular array of numbers arranged in rows and columns.

Example: A 2x2 matrix looks like this: [[1, 2], [3, 4]].

Determinant
A scalar value that can be computed from the elements of a square matrix.

Example: The determinant of [[a, b], [c, d]] is ad - bc.

Linear Equation
An equation that makes a straight line when graphed.

Example: y = 2x + 3 is a linear equation.

Square Matrix
A matrix with the same number of rows and columns.

Example: A 3x3 matrix is a square matrix.

Cramer's Rule
A mathematical theorem used to solve systems of linear equations using determinants.

Example: Cramer's Rule can solve for x in a system of equations.

Row Reduction
A method for simplifying matrices to solve linear equations.

Example: Row reduction can transform a matrix into row echelon form.

Related Topics

Matrix Inversion
The process of finding the matrix that, when multiplied with the original matrix, yields the identity matrix.
intermediate
Eigenvalues and Eigenvectors
Concepts that involve finding scalars and vectors that provide insight into the properties of matrices.
advanced
Linear Transformations
Functions that map vectors to vectors in a linear manner, often represented by matrices.
intermediate

Key Concepts

DeterminantsCramer's RuleLinear EquationsMatrix