| Given that A and B are similar matrices of size n. Then prove the following. i. |A| = |B| ii. AT and BT are similar.

Q: Given that A and B are n×n matrices such that det(A) = 2 and det(B) = −3, find det(A−1BT) ).

A: Given: Matrices A and B are of same order n×n and detA=2, detB=-3 To find: det(A-1BT)

Q: Let B be the matrix obtained from A after the row op- erations 2r3 → r3, r1 ↔ r2, 4r1 + r3 → r3, and…

A: Given: The det(B)=2 The row operations are: 2R3→R3R1↔R24R1+R3→R3-2R1+R4→R4 Let det(A) be x.

Q: Find the elementary matrix E such that EB = A where 2 -12 2 1 1 A = 4 0. -2 and B = 24 %3D | 6. -8 3…

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Q: Prove the formula for a nonsingular n × n matrix A. Assume n ≥ 2.adj[adj(A)] = ∣A∣n−2A

A: We  start with the fundamental equation: A.adj(A) = |A|In

Q: Find the elementary matrix E such that EB = A where %3D 5 4 1 ] 4 1 A = |4 |7 -6 1 -1 and B = |12 3…

A: EB=A multiplying with B-1 on both sidesEBB-1=AB-1E=AB-1

Q: Let A and B be m × n matrices that are row equivalent. (a) Prove that rank  A = rank B . (b) Prove…

A: Given :  A and B are m×n matrices with row equivalentNull Space:…

Q: Find the elementary matrix E such that EC = A where 2 1 5 2 1 A = 3 0 -1 and C- 3 Communest 0 6 -8 2…

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Q: Assume A and B are invertible n-by-n matrices. Which of the following identities is false? O det AT…

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Q: In the given question find an elementary matrix E such that B=EA.

A: given,       A=21-13  ,  B=-1321

Q: Thich of the following is a non-singular matrix P such that PAP is diagonal if 7 0 ? -4 2 A =

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Q: Let A and B be n × m and m × n matrices respectively. If AB = In and BA = Im prove let m = n. Where…

A: Invertible  matrix problem....

Q: Let A and B be n × n matrices such that AB = I. Prove that ∣A∣ ≠ 0 and ∣B∣ ≠ 0.

A: Since AB = I, therefore determinant of AB is 1, that is |AB|=1. This implies that AB is invertible.…

Q: For each of the following pairs of matrices, find an elementary matrix E such that AE = B

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Q: If A is an n×n matrix with the property that Ax = 0 for all x ∈ Rn, show that A = O. Hint: Let x =…

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Q: Prove that if A and B are similar matrices, then ∣A∣ = ∣B∣. Is the converse true?

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Q: Let A and B be two matrices of size 3 x 3 such that det(A) 2 and det(B) 3. Then det(3B- AT) = I| 18…

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Q: Let A and B be square matrices of order n satisfying Ax = Bx for all x in Rn. (a) Find the rank and…

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Q: Let A = Anxn be an invertible matrix. Which of the following statements is not true? %3D O There…

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Q: Find elementary matrices e and e2 such that di 0 e2 = 1 5 1 5 e1 0 dz) for some numbers d and d2.

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Q: Explain why each of the following matrices A is unitarily diagonalizable, without unitarily…

A: Hi! As per norms, we will be answering only the first one. If you need an answer to others then…

Q: Which of the following matrices is a nilpotent matrix of degree 2. a a) where ab =-x². -x a where ab…

A: We have to select the nilpotent matrices of order 2 from the above list.

Q: Let A and B be two matrices of size 3 x 3 such that det(A) =-2 and det(B) = 1. %3D Then det(2A¯²B")…

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Q: Prove that, for m X n matrices A and B, rank (A + B) <=rank(A) + rank(B

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Q: in) For each of the following matrices, determine the value of the scalar k that makes the matrix…

A: The objective is to determine the value of the scalar k such that the given matrices are defective.…

Q: Consider the following matrix: 1 a A = -1 1 -1 Find the valuc of a E R such that A is…

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Q: Decide whether the following statement is true or false. Here A is an n x n matrix. If the equation…

A: The equation Ax=0 has only the trivial solution then columns of A form a linearly independent set.…

Q: Let A and B be invertible n×n matrices. Show that det(A)=det(B) if and only if ?=??A=UB where U is a…

A: This question is related to matrix, we will solve it , we will solve it using given information.

Q: Suppose that A and B are invertible n x n matrices. Which of the following statements is true. A.…

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Q: Prove: If A and B are m × n matrices such that Ax = Bx for all x ∈ R n , then A = B.

A: Given, If A and B are m×n matrices such that Ax=Bx for all x∈Rn Prove that: A=B

Q: 1. Prove the following for any two matrices A and B of order ( 2 × 2 ). (A + B)² = A² + AB + BA + B2…

A: Here (a) and (b) are unrelated problems. These 2 are unrelated problems as anyone can be solve…

Q: let A and B be n ×n invertible matrices. Prove that AB invertible and (AB)−1 = B−1A−1.

A: Let A and B be n×n invertible matrices, with inverses A–1 and B–1, respectively. Consider their…

Q: Let A and B be two matrices of size 3 x 3 such that det(A) = -2 and det(B) = 3. Then det(2A-'B") =…

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Q: 7. The rank of an identity matrix of order n is: А. п - 1 В. п C. n+1 D. 0

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Q: (d) Suppose A is a 2 × 2 matrix such that for all x, y E R?, (Ax) · (Ay) = x · y Must it be true…

A: This is the problem of linear transformation.

Q: Let A and B be nx n matrices. Which of the following statements are always true? (i) (AB)² = 4²B?

A: i   AB2=A2B2-Not Trueii   Ax=0 has infinity many solution then A is not inversible .…

Q: Let A and C be n×n matrices such that ​CA=I​ (the n×n identity​ matrix). Show that Ax​=0 has only…

A: Here, CA=I, where I is the identity matrix.

Q: For a matrix A having JA| = 0, which of the following are true? * O A- exists with |A¬' |= 0| O A-…

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Q: If A and B are square matrices of the same size such that det (A) = det(B), then det(A+ B) =…

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Q: Let A. B. C be m x n matrices. Which of the following is NOT always true? O ABTand BT A have the…

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Q: •(b) A = B:

A: Given,        A=-1201,   B=1-201

Q: Let A and B be two matrices of size 3 x 3 such that det(A) = -2 and det(B) = 3. Then det(2A-1B") =…

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Q: 4. Perform the following operations for the given matrices. a. [E] + [B] b. [A] x [F] (A] x c. [B] -…

A: The given matrices are,A=471256, B=437127204, F=301173, E=158723406

Q: Let B be a Sx5 matrix and Caa): (a-1)(X-メーい(メ) Than the rann of 21-B is? A)S B) 4 c) 3 D) can't…

A: Rank of any matrix is equal to number of non zero rows of matrix when it is in row reduced echelon…

Q: (8) Let A and B be m x n matrices. Prove that the rank of A + B is less than or equal to the rank of…

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Q: For every square matrix A and every scalar c, we have det(cA) = c det(A). O True O False

A: Given statement is  For every square matrix A and every scalar c , we have det ( cA ) = c det ( A )…

Q: Let A and B be 0 1 -3 1 2 A = 1 2 B = 0 1 -3 2 2 -1 2 2 -1 | Find an elementary matrix E such that…

A: When we perform a single row operation on this identity matrix we get a matrix known as the…

Q: Select all of the following that are true for all nxn matrices A and B. If AB=0 then A=0 or B=0.…

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Q: For each of the following symmetric matrices A, find a matrix Q such that Q- = Q" and Q'AQ is…

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Q: A

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Q: Let A be the n × n matrix given by Show that det (A) = (a - 1) n − 1(a + n - 1)

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