Question 5. Suppose that a square matrix X satisfies X3 = X. Do not assume that...
3. (20 %) If a square matrix A satisfies (20201 – A)3 = 0, is it possible that A is not invertible? Explain your answer.
3.23 True or false. justify your answer 190 LINEAR TRANSFORMATIONS 3.22 Let A be a 4 x 3 matrix and B a 3 x 4 matrix. Then AB cannot be in 3.23 Suppose that A is an invertible matrix and B is any matrix for which BA i 3.24 Suppose that A is an invertible matrix and B is any matrix for which AB is 3.25 Suppose that A and B are nxn matrices such that AB is invertible. Then...
5 points 1. True of False: a. if A is an n x1 matrix and B is a 1 xn matrix, then AB is an n xn matrix. b. if A is an n x1 matrix and B is a 1 x n matrix, then BA is not defined. 20 points 2. Use the Invertible Matrix Theorem to determine which of the matrices below are invert- ible. Use as few calculations as possible. Justify your answers. [34 01 4 5...
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
5. Find a 2 x 2 matrix A such that A2 = I2, but A + +12. (Hint: you can do this algebraically, or geometrically.) For all the remaining questions, let n > 2 and let A and B be n x n matrices. 6. Does the equation A(B – In) + (In – B)A = On,n always hold? Either prove it or give a counter-example. 7. If A and B are invertible, does that imply that AB is invertible?...
Suppose A is a square matrix such that det A4 invertible. 0. Prove that A is not Suppose that A is a square matrix such that det A" invertible and that it must have determinant 1. 1. Prove that A is Matrices whose determinant is 1 are part of a group (not just the english word, a special math term, ask if you want the deets) called the Special Linear Group, denoted SL(n) + Drag and drop your files or...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any XER, we can write A= XI + (A - XI) (b) (10 marks) Suppose V is a proper subspace of Mn.n(R). That is to say, V is a subspace, and V #Mnn(R) (there is some Me M.,n(R) such that M&V). Show that there exists an invertible matrix M e Mn.n(R) such...
(a) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any le R, we can write A = XI + (A - XI) (b) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn.n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M e Mn,n(R) such that M&V....