Exam 3: Matrices

The identity matrix is a regular stochastic matrix.

Find , given that

Use the given LU factorization to solve where

Every matrix has an LU factorization.

If and are doubly stochastic matrices, then AB is a doubly stochastic matrix.

If is a one-to-one linear transformation from R² to R², then the image of the unit square under is a parallelogram with positive area.

If is a square invertible matrix, then is an invertible matrix for every positive integer , with

A matrix is doubly stochastic if and only if is doubly stochastic.

Find all steady-state vectors for the given stochastic matrix.

Determine whether is one-to-one, and whether is onto, where and

Find an LDU factorization of

Use an augmented matrix and row operations to find the inverse of

Use the matrix to find the solutions to the linear system

Evaluate AB-BA, where

Determine if the given vector is in the range of where

If for a matrix , and is an onto linear transformation, then the number of columns of is greater than or equal to the number of rows of .

Find an LU factorization of

If and are invertible matrices, then so is

Find the state vector for the given stochastic matrix and initial state vector.

Solve for the scalars a, b, c in the following equation: