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Gaussian Elimination: derived system back-substitution

Exercise: Solve the system by Gaussian elimination:

The Inverse of a Matrix: A is the inverse of B B is the inverse of A Example:

Example: Matrices with zero rows or columns have no inverses properties of the matrix inverse

An inverse of a product of two and more invertible matrices Example: verify Generalize for a product of any number of invertible matrices:

Exercise: A square matrix A satisfies A² -3A+I = 0. Find the inverse of A.

Invertability of a 2 x 2 Matrix: Exercise: Verify. Exercise: Find A.

Example: an inverse of a diagonal matrix

Elementary Matrices: Examples: Inspect the effect of multiplication involving “special” matrices.

Elementary Matrices: A simple Way to Construct an Elementary Matrix Examples:

Exercise:

Inverses of Elementary Matrices:

Exercise: Find inverses for elementary matrices

Invertability of a Square Matrix:

Computing the Inverse: convert to the identity matrix by row operations

Example: Find the inverse of A.

Why are we interested in finding the matrix inverse? Solving Matrix equations with Inverses: Example: Solve the system using the inverse.

LU-decomposition “A has an LU-decomposition” Theorem A allows an LU-decomposition Theorem

Solving SLE equations using LU-decomposition:

Corresponding elementary matrices

lower-∆ A = L U

Exercise: find an LU-decomposition