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 B  A1 . 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 A  AT .
 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 a ij is the submatrix M ij obtained from A




by deleting the i-th row and the j-th column.
Cofactor of an entry: the cofactor of an entry a ij is the C ij = (1) i  j det( M ij ).
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 Ax  b has a unique
solution if and only if the inverse of the coefficient matrix A exists. The solution
is given by x  A 1b .
 The following statements are equivalent for an nxn matrix A:
o inverse A exists
o A is row equivalent to the identity matrix
o det( A)  0
o Ax  b has a unique solution
Cramer’s Rule: The solution of a system Ax  b where A is invertible (nonsingular) is
det( Ai )
given by xi =
, where Ai is the matrix obtained by replacing the i-th column of A
det( A)
with b.
LU-decomposition and systems of linear equations Ax  b : If A admits such an LUdecomposition, then one can solve two triangular sparse systems Ly  b using forward
elimination and Ux  y using back substitution.
Review Questions:
1. Give an example of two square matrices to show that ( A  B) 1  A 1  B 1 .
2. If the system Ax  b has infinitely many solutions, does the inverse of the matrix A
exist?
3. If A is a 3x3 matrix whose det( A)  2 , find:
det(3A) ; det( A3 );
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 Ax  b .
5. For what values of x does the matrix:
 x 1  1
0 x 2 


0 4 x 
have an inverse?
6. Find the cofactor matrix of the matrix:
 2 1  1
1 3 4


 1 4 2 
7. Find the adjoint matrix of the matrix A:
 2 1  1
1 3 4


 1 4 2 
and deduce the inverse of the matrix.
8. Find the LU decomposition of the matrix A:
1 1
1
0
1 4

 1  2 2
 2
Use this decomposition to solve the system Ax  b where b  6 .
8 