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HomeHomework HelpmathematicsMatrix Inversion

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix. It is a fundamental concept in linear algebra, widely used in solving systems of linear equations.

intermediate
2 hours
Mathematics
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Overview

Matrix inversion is a fundamental concept in linear algebra that allows us to solve systems of linear equations and perform various transformations. By finding the inverse of a matrix, we can effectively 'undo' the effects of that matrix, leading to solutions in many practical applications, such as ...

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Key Terms

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

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

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

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

Determinant
A scalar value that can be computed from the elements of a square matrix, indicating if it is invertible.

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

Adjugate Matrix
The transpose of the cofactor matrix, used to find the inverse of a matrix.

Example: For A = [[a, b], [c, d]], adj(A) = [[d, -b], [-c, a]]

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

Example: If A = [[1, 2], [3, 4]], then A⁻¹ = [[-2, 1], [1.5, -0.5]]

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

Example: A 2x2 matrix is square.

Related Topics

Eigenvalues and Eigenvectors
Study the properties of eigenvalues and eigenvectors, which are crucial in understanding matrix transformations.
intermediate
Linear Algebra
Explore the broader field of linear algebra, which includes vector spaces and linear transformations.
advanced
Systems of Linear Equations
Learn how to solve systems of linear equations using matrices and their inverses.
intermediate

Key Concepts

Identity MatrixDeterminantAdjugate MatrixInverse Matrix