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HomeHomework HelpmathematicsMatrix Inverses

Matrix Inverses

Linear algebra is a branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. The concept of matrix inverses is critical in solving systems of linear equations, where a matrix is invertible if it has a unique solution.

intermediate
3 hours
Mathematics
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Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. Understanding matrix inverses is crucial for solving systems of linear equations, as they allow us to find unique solutions. The inverse of a matrix is defined such that when it is multiplied by ...

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Key Terms

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

Example: A 2x2 matrix: [[1, 2], [3, 4]]

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

Example: The 2x2 identity matrix: [[1, 0], [0, 1]]

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

Example: For matrix [[a, b], [c, d]], the determinant is ad - bc.

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

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

Singular Matrix
A square matrix that does not have an inverse.

Example: A matrix with a determinant of zero is singular.

Row Reduction
A method for simplifying matrices to find their inverses or solutions to systems.

Example: Using Gaussian elimination to row-reduce a matrix.

Related Topics

Eigenvalues and Eigenvectors
Study how matrices can stretch or shrink vectors, crucial for understanding transformations.
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Linear Transformations
Explore how matrices can represent transformations in space, linking to matrix inverses.
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Systems of Linear Equations
Learn how to solve systems using matrix methods, including the use of inverses.
intermediate

Key Concepts

Matrix MultiplicationIdentity MatrixDeterminantsInverse Calculation