Solution: In Exercises 1316, show that the given matrix is positive definite,and then

In Exercises 16, find the principal submatrices of the given matrix. A = 3 5 5 7

In Exercises 16, find the principal submatrices of the given matrix. A = 1 6 6 2 3

In Exercises 16, find the principal submatrices of the given matrix. A =  1 4 3 40 2 32 5

In Exercises 16, find the principal submatrices of the given matrix. A = 302 0 4 1 2 1 1

In Exercises 16, find the principal submatrices of the given matrix. A = 21 0 1 13 4 1 04 0 2 1 1 2 1

In Exercises 16, find the principal submatrices of the given matrix. A = 1 23 0 2 42 3 3 23 1 0 31 0

In Exercises 712, determine if the given matrix is positive definite. A = 2 1 1 2

In Exercises 712, determine if the given matrix is positive definite. A = 1 3 3 -5

In Exercises 712, determine if the given matrix is positive definite. A =  1 2 1 25 1 1 1 11

In Exercises 712, determine if the given matrix is positive definite. A =  1 3 2 3 10 7 2 7 6

In Exercises 712, determine if the given matrix is positive definite. A = 1121 1210 2163 1036

In Exercises 712, determine if the given matrix is positive definite. A = 1 221 2 5 1 3 2 1 14 1 1 3 1 3

In Exercises 1316, show that the given matrix is positive definite, and then find the LU-factorization A = 1 2 2 5

In Exercises 1316, show that the given matrix is positive definite, and then find the LU-factorization A = 1 3 3 10 1

In Exercises 1316, show that the given matrix is positive definite, and then find the LU-factorization A =  1 2 2 2 5 5 2 5 6

In Exercises 1316, show that the given matrix is positive definite, and then find the LU-factorization A =  1 1 1 1 10 4 1 4 11

In Exercises 1720, show that the given matrix is positive definite, and then find the LDU-factorization. A = 1 2 2 8

In Exercises 1720, show that the given matrix is positive definite, and then find the LDU-factorization. A = 4 4 4 5

In Exercises 1720, show that the given matrix is positive definite, and then find the LDU-factorization. A = 1 32 3 10 8 2 89

In Exercises 1720, show that the given matrix is positive definite, and then find the LDU-factorization. A = 111 151 115

In Exercises 2124, show that the given matrix is positive definite, and then find the Cholesky decomposition. A = 1 2 2 5

In Exercises 2124, show that the given matrix is positive definite, and then find the Cholesky decomposition. A = 4 2 2 10

In Exercises 2124, show that the given matrix is positive definite, and then find the Cholesky decomposition. A = 12 3 25 6 3 6 10

In Exercises 2124, show that the given matrix is positive definite, and then find the Cholesky decomposition. A = 141 4 25 7 176

FIND AN EXAMPLE For Exercises 2530, find an example that meets the given specifications. A 2 2 matrix A such that det(A1) < 0 but det(A2) > 0.

FIND AN EXAMPLE For Exercises 2530, find an example that meets the given specifications. A 2 2 matrix A such that det(A1) > 0 but det(A2) < 0.

FIND AN EXAMPLE For Exercises 2530, find an example that meets the given specifications. A 33 matrix A such that det(A1) < 0 and det(A3) < 0, but det(A2) > 0.

FIND AN EXAMPLE For Exercises 2530, find an example that meets the given specifications. A 3 3 matrix A such that det(A1) > 0, but det(A2) < 0, and det(A3) < 0.

FIND AN EXAMPLE For Exercises 2530, find an example that meets the given specifications. A 3 3 matrix A such that det(A) > 0 but A is not positive definite.

FIND AN EXAMPLE For Exercises 2530, find an example that meets the given specifications. A 3 3 matrix A such that det(A) < 0 but A is not negative definite.

TRUE OR FALSE For Exercises 3136, determine if the statement is true or false, and justify your answer. A symmetric matrix A is positive definite if and only if det(A) > 0.

TRUE OR FALSE For Exercises 3136, determine if the statement is true or false, and justify your answer. If Lc L T c is the Cholesky decomposition of A, then L T c Lc is the Cholesky decomposition of AT .

TRUE OR FALSE For Exercises 3136, determine if the statement is true or false, and justify your answer. If A and B are n n positive definite matrices, then so is AB.

TRUE OR FALSE For Exercises 3136, determine if the statement is true or false, and justify your answer. If A and B are n n positive definite matrices, then so is A + B.

TRUE OR FALSE For Exercises 3136, determine if the statement is true or false, and justify your answer. If A is a positive definite matrix, then so is A1.

TRUE OR FALSE For Exercises 3136, determine if the statement is true or false, and justify your answer. If the determinants of the leading principal submatrices of a symmetric matrix A are all negative, then A is negative definite.

Let A = L1D1U1 and A = L2D2U2 be two LDUfactorizations of A. Prove that L1 = L2, D1 = D2, and U1 = U2, which shows that the LDU factorization is unique.