MATH 225N – Applied Statistics for Nursing Exam Study Guide with Questions & Answers

Math 225 Final Exam

If the columns of A are linearly dependent - ANSWER Then the matrix is not invertible and an
eigenvalue is 0



Note that A−1 exists. In order for λ−1 to be an eigenvalue of A−1, there must exist a nonzero x
such that Upper A Superscript negative 1 Baseline Bold x equals lambda Superscript negative 1
Baseline Bold x . A−1x=λ−1x. Suppose a nonzero x satisfies Ax=λx. What is the first operation
that should be performed on Ax=λx so that an equation similar to the one in the previous step
can be obtained? - ANSWER Left-multiply both sides of Ax=λx by A−1.



Show that if A2 is the zero matrix, then the only eigenvalue of A is 0. - ANSWER If Ax=λx for
some x≠0, then 0x=A2x=A(Ax)=A(λx)=λAx=λ2x=0. Since x is nonzero, λ must be zero. Thus, each
eigenvalue of A is zero.



Finding the characteristic polynomial of a 3 x 3 matrix - ANSWER Add the first two columns to
the right side of the matrix and then add the down diagonals and subtract the up diagonals



In a simplified n x n matrix the Eigenvalues are - ANSWER The values of the main diagonal



Use a property of determinants to show that A and AT have the same characteristic polynomial -
ANSWER Start with detAT−λI)=detAT−λI)=det(A−λI)T. Then use the formula det AT=det A.



The determinant of A is the product of the diagonal entries in A. Select the correct choice below
and, if necessary, fill in the answer box to complete your choice. - ANSWER The statement is
false because the determinant of the

2×2 matrix A= [ 1 1 (1 1 below) ] is not equal to the product of the entries on the main diagonal
of A.



An elementary row operation on A does not change the determinant. Choose the correct
answer below. - ANSWER The statement is false because scaling a row also scales the
determinant by the same scalar factor.

, (det A)(det B)=detAB. Select the correct choice below and, if necessary, fill in the answer box to
complete your choice. - ANSWER The statement is true because it is the Multiplicative
Property of determinants.



If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. Select the
correct choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWER The statement is false because in order for 5 to be an eigenvalue of A, the
characteristic polynomial would need to have a factor of λ−5.



Determine whether the statement "If A is 3×3, with columns a1, a2, a3, then det A equals the
volume of the parallelepiped determined by a1, a2, a3" is true or false. Choose the correct
answer below. - ANSWER The statement is false because det A equals the volume of the
parallelepiped determined by a1, a2, a3. It is possible that det A≠det A.



Determine whether the statement "det AT=(−1)det A"is true or false. Choose the correct answer
below. - ANSWER The statement is false because det AT=det A for any n×n matrix A.



Determine whether the statement "The multiplicity of a root r of the characteristic equation of
A is called the algebraic multiplicity of r as an eigenvalue of A" is true or false. Choose the
correct answer below. - ANSWER The statement is true because it is the definition of the
algebraic multiplicity of an eigenvalue of A.



Determine whether the statement "A row replacement operation on A does not change the
eigenvalues" is true or false. Choose the correct answer below. - ANSWER The statement is
false because row operations on a matrix usually change its eigenvalues.



A matrix A is diagonalizable if A has n eigenvectors. - ANSWER The statement is false. A
diagonalizable matrix must have n linearly independent eigenvectors.