Problem 33. A matrix A € Mnxn (F) is called nilpotent if, for some positive integer k, Ak = A... A = 0, n times. where O is the n x n zero matrix. Prove that if A is nilpotent, then det(A) = 0.

Q: 2- For any constant matrix A, if A² = 0 then a- e4' =I +At b- e4 =I - A12 c- e At-A²1 ² The correct…

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Q: Suppose A is any nilpotent matrix (that is, for some n > 1, A" = Onxn, the nxn zero matrix). Find…

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Q: justifying your answers: 10 2 a) Let A = [1 0 k]and B =-1 2 -1. If ABAT = 0, then k = 1. b) If p(x)…

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Q: 8. Give an example of a nonzero matrix A such that that Aª = 0 for a positive integer n. Show that…

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Q: #12. Let A be an m × n matrix and B be an n × p matrix. Prove that (a) N(B) C N(AB). (b) C(AB) C…

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Q: 2. (a) Let S= A 0 I B A, B & GL (R), B = B, A = (B n ¸ Check whether or not S SP2n(R). Also give an…

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Q: 5) A matrix A is diagonal if A¡j = 0 for i + j. Entries on the diagonal are not required to be…

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Q: 1. Find the general solution of x =x2 – x1 + t, x, =x2. 1 2 3 2. Explain why we cannot aclculate the…

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Q: 11. Any square matrix A with det(A)=-2 has an inverse matrix. True False

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: ^ - [ 3 ] and B = [² = 4 5 If A 4 5 7 6 find 3A-2B+ I where I is a unit matrix.

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Q: If an n × n matrix A satisfies A? – 6A + 5In = 0, then O is (1/5)(6In – A). O is (1/5)(A – 6In). O…

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Q: 3. Suppose we have a matrix A = 1 1 101 1 2 2 What is rank of A?

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Q: Given the matrix A || [1 (a) Find the value of x that makes A singular. (b) Find the value of x that…

A: Given matrix is  A = -14x1 (a)  If the matrix A is singular then   det (A)  = 0…

Q: %3D For every matrix A, it is true that (A' )' = A. O True O False

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Q: 2 4 8 Suppose that A = LDU = 039 , then the matrix U is 0 0 7 equal to: 2 00 0 3 0 0 0 7 1 2 4 0 1 3…

A: Given:Suppose A=LDU=248039007To find : Matrix U

Q: 7. For a matrix B= 1 0 0 1 -1 0 5-1 -3 find B¹ by using the Cayley-Hamilton theorem.

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Q: .2) x(n)= n2+10 LTI h(n) ho, hỗ y(n)=10 1- By using 2x2 square input matrix, find h, and hy 2- Find…

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Q: suppose that A is an nxn matrix It A² = A then A must be the identity. Cuote assume that no1).

A: is an  matrix.Also, .

Q: 4. If A is a 4x4matrix and det A=2 , then det(34)is equal to (a) (b) 162 (c) 48

A: Given query is to find the det(3A) when matrix A is a 4 × 4 matrix.

Q: Let A be a 3 x 10 matrix, and let B be m x n. Suppose that B'A' is a 21 x 3 matrix. What is the…

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Q: 2A-1-630- 1 3. If A= 3 B= -5 and D= 4 Γα [] Le , find a matrix C= c d such that 3A + B - 2C= D.

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Q: A number is very close to zero. Another number is also very close to zero. The quotient, , therefore…

A: Given a number is very close to zero. Another number is also very close to zero. Let α>0 is very…

Q: Assume B is an n × n matrix, n > 1, and det(B) = 4. (a) Find n such that det(−5B) = 100

A: Given:We are required to find:

Q: Problem 1. (a Let A = (" ), (a, b, c, and d are real numbers.) (a) Show that A is positive definite…

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Q: If A is n x n matrix and (I- A)² = I- A then O a. AT = A %3D O b. AT A = I O c. AAT = I Od. A? = A

A: We have to solve given problem:

Q: 4. If A is a nonsingular nxn matrix, show that AA¹ is nonsingular and det(AA) > 0.

A: Given A is a nonsingular matrix.

Q: 1. If A and B are different rank-one matrices ( A* B), then the rank of A-B is a =2 b. d 21 c. If…

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Q: 5. Consider the matrix 1 3 5 A = 18 18 18 5 3 1 Is A invertible? Justify your answer.

A: Given, A=135181818531A is invertible if and only if its determinantA≠0.

Q: 1. Find using Method of Undetermined Coefficient (MUC): y(3) + y(2) + y(a) + y = 2e-2* – 4x° + cos…

A: 1. Given : differential equation         y3+y2+y1+y=2e-2x-4x2+cos3x, y0=-3, y'0=-1, y''0=2…

Q: Problem 7. Let A be an nxn matrix. (a) Prove that if A is singular, then adi A must also be…

A: # as per the guidelines we are entitled to solve one question at a time ,please resubmit the other…

Q: Suppose K is the 9 by 9 second difference matrix (2's on the diagonal, -1 's on the diagonal above…

A: As per the question we are given the 9×9 second difference matrix K (2's on the diagonal, -1 's on…