![I side Aus AXAT = I - take determinant both =) LAXAT] = ca and (Ile) - _LAB)= JAH]B)__ JALAT = 1 JAL=1A - IAP= =)/ 1912 EJ So](http://img.homeworklib.com/questions/ad401610-9eeb-11eb-98e3-c7731b8b02e6.png?x-oss-process=image/resize,w_560)
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
2. Let A be an n x n matrix with AT =-A (a) Prove that A has value 0. (b) Prove that A has determinant 0 if n is odd.
Let A be an ( n x n ) matrix, and let Lambda be an eigenvalue of A. Prove that for any scalar Alpha, Lambda + Alpha is an eigenvalue of A + Alpha x I (identity matrix).
[3] 7. Let A be a square matrix such that A# I and A+ -I with eigenvalue X. Prove that if AP = I(I is the identity matrix), then = 1 or = -1.
How do you do this Linear Algebra problem?
6. Let A [ai i be an mxn matrix with RREF R-FF. Prove that i.. Tn there exists an m × m invertible matrix E such that аґ Eri for 1-i-n
6. Let A [ai i be an mxn matrix with RREF R-FF. Prove that i.. Tn there exists an m × m invertible matrix E such that аґ Eri for 1-i-n
(3) 7. Let A be a square matrix such that A# I and A+ - with eigenvalue A. Prove that if AP = (is the identity matrix), then X = 1 or X = -1.
*Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1
*Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cofactor expansion across the ith row of A, that is, det A H-1)adtAj Hint: Use induction on i, For the induction step from i to i+1, flip rows i and i+1 (How does this change the determinant?) and use the induction assumption.
1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent.
Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent.