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HomeHomework Helplinear-algebraMatrix Invertibility

Matrix Invertibility

Matrix invertibility refers to the condition under which a square matrix has an inverse, which is equivalent to the matrix being row-equivalent to the identity matrix through a series of elementary row operations.

intermediate
2 hours
Linear Algebra
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Overview

Matrix invertibility is a fundamental concept in linear algebra that determines whether a matrix can be inverted to solve linear equations. A matrix is invertible if its determinant is non-zero, which indicates that it has full rank. Understanding how to manipulate matrices through row operations, s...

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Key Terms

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

Example: A = [[1, 2], [3, 4]]

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

Example: det(A) = ad - bc for A = [[a, b], [c, d]]

Row Echelon Form
A form of a matrix where all non-zero rows are above any rows of all zeros.

Example: For [[1, 2], [0, 1]], it is in row echelon form.

Gaussian Elimination
A method for solving systems of linear equations by transforming the matrix to row echelon form.

Example: Used to find the solution of Ax = b.

Inverse Matrix
A matrix that, when multiplied by the original matrix, yields the identity matrix.

Example: If A is invertible, then A * A^(-1) = I.

Identity Matrix
A square matrix with ones on the diagonal and zeros elsewhere.

Example: I = [[1, 0], [0, 1]]

Related Topics

Linear Transformations
Study how matrices can represent transformations in space, including rotations and scaling.
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Eigenvalues and Eigenvectors
Explore the concepts of eigenvalues and eigenvectors, which are critical in understanding matrix behavior.
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Systems of Linear Equations
Learn how to solve systems of linear equations using various methods, including substitution and elimination.
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Key Concepts

Invertible MatrixRow Echelon FormGaussian EliminationDeterminant