MATH225 EXAM 2026 STATISTICS
PROBABILITY QUESTIONS SOLUTIONS
GRADED A+
◉ 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.
PROBABILITY QUESTIONS SOLUTIONS
GRADED A+
◉ 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.
