Unit Two Review
Matrix Algebra
Review of terms:
· Nonsingular matrix: an nxn matrix A such that there is an nxn matrix B for which AB = BA = I where I is the identity matrix. B is called the inverse of the matrix A and is usually denoted by . Another equivalent term is an invertible matrix.
· Singular matrix: an nxn matrix A that does not have an inverse; that is, you cannot find an nxn matrix for which AB = BA = I where I is the identity matrix.
· Transpose of a matrix: The transpose of a matrix A is a matrix B whose rows are the columns of the matrix A.
· Symmetric matrix: an nxn matrix A for which .
· Elementary matrix: an nxn matrix which is obtained from the identity matrix by the application of one elementary operation.
· Upper (lower) triangular matrix: an matrix for which the entries below (above) the diagonal are equal to zero.
· Minor of an entry: the minor of an entry is the submatrix obtained from A by deleting the i-th row and the j-th column.
· Cofactor of an entry: the cofactor of an entry is the =).
· Cofactor matrix: an matrix obtained from a given matrix A by replacing each entry by its corresponding cofactor.
· Adjoint of a matrix A: an nxn matrix which is equal to the transpose of the cofactor matrix.
· LU-decomposition: an algorithm by which one can write a given matrix A as the product of a lower triangular and an upper triangular matrices.
Review of some facts:
· The inverse of a matrix is unique.
· Solution set of a system of linear equations: the system has a unique solution if and only if the inverse of the coefficient matrix A exists. The solution is given by .
· The following statements are equivalent for an nxn matrix A:
o inverse A exists
o A is row equivalent to the identity matrix
o
o has a unique solution
Cramer’s Rule: The solution of a systemwhere A is invertible (nonsingular) is given by = , where is the matrix obtained by replacing the i-th column of A with b.
LU-decomposition and systems of linear equations: If A admits such an LU-
decomposition, then one can solve two triangular sparse systems using forward elimination and using back substitution.
Review Questions:
1. Give an example of two square matrices to show that.
2. If the systemhas infinitely many solutions, does the inverse of the matrix A exist?
3. If A is a 3x3 matrix whose, find:
det(3A) ; det();
det (Adj(A)), where Adj(A) is the adjoint matrix corresponding to A; and
det(C), where C is the cofactor matrix corresponding to A.
4. If a matrix A is row equivalent to the identity matrix, describe the solution set of the system.
5. For what values of x does the matrix:
have an inverse?
6. Find the cofactor matrix of the matrix:
7. Find the adjoint matrix of the matrix A:
and deduce the inverse of the matrix.
8. Find the LU decomposition of the matrix A:
Use this decomposition to solve the system where .