a) If A and B are invertible n n matrices, then so is A + B.b) The rank of a 4 5 matrix may be 5.c) The rank of a 5 4 matrix may be 5.d) If S and A are invertible, then (S?1AS)?1 = S?1A?1S.e) If A and B are given n n matrices, then there is a unique n n matrix X satisfying (A+X)B = Aif B invertible.f) It is possible that a system Ax = b has a unique solution for some b if A is a 4 5 matrix.g) If A is a 5 5 matrix such that the system Ax = 0 has only the trivial solution, then the systemAx = b is consistent for every b 2 R5.h) An n n matrix of rank n is invertible.i) The system Ax = 0 has only the trivial solution if and only if there are no free variables.j) For any two n n matrices A and B, (A + B)2 = A2 + 2AB + B2.k) For a 4 4 matrix A, det(3A)=3det(A).Which of the statements above are true? 3)a) If A and B are invertible n x n matrices, then so is A + B.b) The rank of a 4 x 5 matrix may be 5.c) The rank of a 5 x 4 matrix may be 5.d) If S and A are invertible, then (S-'AS)-1 = s-1A-1s.e) If A and B are given n xn matrices, then there is a unique n x n matrix X satisfying (A +X)B = Aif B invertible.f) It is possible that a system Ax = b has a unique solution for some b if A is a 4 x 5 matrix.g) If A is a 5 x 5 matrix such that the system Ax = 0 has only the trivial solution, then the systemAx = b is consistent for every b e R*.h) An n xn matrix of rank n is invertible.i) The system Ax = 0 has only the trivial solution if and only if there are no free variables.j) For any two n xn matrices A and B, (A + B)? - A? + 2AB + B?.k) For a 4 x 4 matrix A, det(3A)-3det (A).Which of the statements above are true?(Submit the corresponding number without parentheses.)(1) b, c, d, f, g.(2) a, b, d, i, k.(3) a, c, d, e, h.(4) d, e, g, h, i.(5) b, c, d, g, h, j.a, c, d, f, g, k.a, b, d, h, j, k.(8) a, b, c, e, g, h, k.a, c, d, e, h, i, j.(10) b, d, e, f, h, j, k.

If a matrix has full rank, then the matrix is said to be invertible.

Answered: a) If A and B are invertible n x n…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

a) If A and B are invertible n n matrices, then so is A + B.
b) The rank of a 4 5 matrix may be 5.
c) The rank of a 5 4 matrix may be 5.
d) If S and A are invertible, then (S?1AS)?1 = S?1A?1S.
e) If A and B are given n n matrices, then there is a unique n n matrix X satisfying (A+X)B = A
if B invertible.
f) It is possible that a system Ax = b has a unique solution for some b if A is a 4 5 matrix.
g) If A is a 5 5 matrix such that the system Ax = 0 has only the trivial solution, then the system
Ax = b is consistent for every b 2 R5.
h) An n n matrix of rank n is invertible.
i) The system Ax = 0 has only the trivial solution if and only if there are no free variables.
j) For any two n n matrices A and B, (A + B)2 = A2 + 2AB + B2.
k) For a 4 4 matrix A, det(3A)=3det(A).
Which of the statements above are true?

3)
a) If A and B are invertible n x n matrices, then so is A + B.
b) The rank of a 4 x 5 matrix may be 5.
c) The rank of a 5 x 4 matrix may be 5.
d) If S and A are invertible, then (S-'AS)-1 = s-1A-1s.
e) If A and B are given n xn matrices, then there is a unique n x n matrix X satisfying (A +X)B = A
if B invertible.
f) It is possible that a system Ax = b has a unique solution for some b if A is a 4 x 5 matrix.
g) If A is a 5 x 5 matrix such that the system Ax = 0 has only the trivial solution, then the system
Ax = b is consistent for every b e R*.
h) An n xn matrix of rank n is invertible.
i) The system Ax = 0 has only the trivial solution if and only if there are no free variables.
j) For any two n xn matrices A and B, (A + B)? - A? + 2AB + B?.
k) For a 4 x 4 matrix A, det(3A)-3det (A).
Which of the statements above are true?
(Submit the corresponding number without parentheses.)
(1) b, c, d, f, g.
(2) a, b, d, i, k.
(3) a, c, d, e, h.
(4) d, e, g, h, i.
(5) b, c, d, g, h, j.
a, c, d, f, g, k.
a, b, d, h, j, k.
(8) a, b, c, e, g, h, k.
a, c, d, e, h, i, j.
(10) b, d, e, f, h, j, k.

Expert Solution

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Determinant$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions