Both problems (1 point) Solve for X. ]+sx= [=] X = (1 point) Let A be...
SOLVE BOTH 4 and 5!!
4. Let A and B be two nxn matrices. Suppose that AB is invertible. Show that the system Ar 0 has only the trivial solution 5. Given that B and D are invertible matrices of orders n and p respectively, and A = Find A by writing A as a suitably partitioned matrix
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
(1 point) Suppose that A, B, and Care 5 x 5,5 X 6, and 6 x 9 matrices, respectively. Determine which of the following products are defined. (a) BC Answer: undefined (b) CB Answer: undefined (c) AB Answer: 5x5 (d) 42 Answer: 5x5 Note: 1) For those defined, enter the size of the resulting matrix (e.g. 3 x 4 with spaces between the numbers and x). ii)For those undefined, enter undefined.
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...
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
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?...
Differention Equations - Can someone answer the checked
numbers please?
Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
Let A be an m x n matrix and let B be an n x p matrix. (a) Prove that Col(AB) SColA) (b) Use part (a) to prove that the rank of AB is at most the rank of A (c) Use transpose matrices to prove that the rank of AB is also at most the rank of B.
Question 5 True of False part II: 5 problems, 2 points each. (6). Let w be the x-y plain of R3, then wlis any line that is orthogonal to w. (Select) (7). Let A be a 3 x 3 non-invertible matrix. If Ahas eigenvalues 1 and 2, then A is diagonalizable. Sele (8). If an x n matrix A is diagonalizable, then n eigenvectors of A form a basis of " [Select] (9). Letzbean x 1 vector. Then all matrices...
Q3 (3 points) Show that if both AB and B A are defined then AB and BA are square matrices. + Drag and drop your images or click to browse... Q4 (3 points) Let A = (a) be a 2 x 2 matrix. The trace of A. which we denote by tr(A) is a number defined as tr(A) = 0 + 0x2. Prove the following properties of this number for 2 x 2 matrices A and B and a real...