Ch 2.2 - The Inverse of a Matrix

Flashcards covering key definitions and theorems related to the inverse of a matrix from the lecture notes '2.2 The Inverse of a Matrix'.

Invertible Matrix

An n x n matrix A for which there exists an n x n matrix C such that CA = I and AC = I, where I is the n x n identity matrix. The unique inverse is denoted by A^-1.

Singular Matrix

A matrix that is not invertible.

Nonsingular Matrix

Another term for an invertible matrix.

Determinant of a 2x2 Matrix

For a matrix A = [[a, b], [c, d]], the determinant is the quantity ad - bc, denoted det A. A 2x2 matrix is invertible if and only if its determinant is not zero.

Theorem 5 (Unique Solution Ax=b)

If A is an invertible n x n matrix, then for each vector b in Rⁿ, the equation Ax = b has the unique solution x = A^-1b.

Flexibility Matrix

In the context of an elastic beam, a matrix D that relates forces (f) to deflections (y) via the equation y = Df.

Stiffness Matrix

The inverse of a flexibility matrix (D^-1), which computes a force vector (f) when a deflection vector (y) is given, such that f = D^-1y.

Properties of Invertible Matrices (Theorem 6)

Describes three properties: (a) If A is invertible, (A^-1)^-1 = A. (b) If A and B are invertible n x n matrices, (AB)^-1 = B^-1A^-1. (c) If A is invertible, (A^T)^-1 = (A^-1)^T.

Elementary Matrix

A matrix obtained by performing a single elementary row operation on an identity matrix. This matrix is always invertible.

Theorem 7 (Invertibility and Row Equivalence)

States that an n x n matrix A is invertible if and only if A is row equivalent to the n x n identity matrix I. Any sequence of elementary row operations reducing A to I will also transform I to A^-1.

Algorithm for Finding A^-1

To find the inverse of A, row reduce the augmented matrix [A I]. If A is row equivalent to I, then [A I] will transform into [I A^-1]; otherwise, A is not invertible.