. Let A and B be 3 X 3 matrices such that det A = 3 and det B = -2. Find the deteminant of  a) AT B b) A2B-1 c) 7A d) (3A)-1

Q: Construct a 3 x 3 triangular matrix A so that 1] the vector 2 is in Col A. 3]

A: We know every lower or upper traingular matrix is a triangular matrix.

Q: Is this the correct definition of upper triangular, for a matrix A? Aij * 0 when j 2 i O Yes No

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

Q: Let A and B be 2x2 matrices such that B is invertible If detA)2 and det)-, then detA'e) Select one:…

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Q: A matrix Q ∈Mn×n(R) is called orthogonal if QQt= I. Prove that if Q is orthogonal, then det(Q) = ±1.

A: The determinant of the product of the matrices is the product of the determinant of the matrices.

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Q: (AB)† = B†A† (AB)−1 = B−1A−1

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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: 2. Show that if A is an n x n real symmetric matrix then (Ax) y = x (Ay) for all x, y E R".

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Q: Let x and y be vectors in Rn and let Q be an n × n orthogonal matrix. Show that if z = Qx and w = Qy…

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Q: Suppose C is the cofactor matrix of matrix A and A CT then O a JA| = 0.5 %3D Ob.A= 1 %3D Od. A=+1

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Q: An n x n matrix A is orthogonal if AA" = I. If A is orthogonal show that det(A) = ±1.

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Let A and B be 5x5 matrices such that det(A) = 4 and det(B) = 3. %3D Compute det(2B)"(AB)")

A: We have to find

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Q: Let A = ; 1.B - : J.c= , ], anc 2 37 ol.c = [ 1 and v = |1. What is the sum of the entries of the L4…

A: Note: We know that for matrix P of order m*n and Q of order n*t, the multiplication PQ is possible…

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Q: Find a matrix A such that W = Col A. W [2s-5t] 2t 2s+t :stin R

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Q: Given the bases B={u , Uz} and B' ={u'ı, u'2} for R?, where 1 Uz= 1 and u' u' 2 %3D The transition…

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Q: Suppose that x and y are mxn matrices. Prove that xT- yT=(X -YT. Find a 2x2matrix A, that is not the…

A: Take A and B as arbitrary m×n matrices and evaluate the left and right sided expression

Q: Prove that for any matrix A ∈ Mm×n(F), (R(LA∗))⊥= N(LA).

A: The objective is to prove that

Q: Let D = P'AP be a diagonal matrix, wher %3D Pesell thet if D:

A: Given, D=P-1AP be a diagonal matrix, where P is an orthogonal matrix. To prove that A is symmetric.

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Q: Show that ||AB||F ≤ ||A||2||B||F and ||AB||F ≤ ||A||F||B||2 for all matrices A and B.

A: The Frobenius norm for a square matrix A is defined as AF=∑i,j=1naij212. The Frobenius norm…

Q: (i) Use the properties of trace to argue that tr(AʼA) = tr(AA') for any n × m matrix A. [2 0 (ii)…

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Q: 1. Let A = [a;j]4x5, B = [bij]4x5; C = [cij]5x2; D = [d:j]4x2, and E which of the following matrix…

A: Matrix addition: Let A be an m×n matrix and B be an also m×n matrix. A= {aij: 1≤i≤m ; 1≤j≤n} and B=…

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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(2A-'B") =…

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Q: Let Bj denote the matrix obtained by replacing thejth column of the identity matrix with a vectorb =…

A: COnsider the system of equations:IX =…

Q: Let B be an n X n matr ix and let E be an n X n elementary matrix. Then det(EB) = (det E)(det B)

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Q: TRUE/FALSE Assume that all the given matrices are n × n. If x is a nonzero vector in Rn and Ax = 0,…

A: Given: All the given matrices are n x n  and  X is a non-zero vector in Rn and AX = 0. To show:…

Q: Consider the ordered bases B = {1+x, 1 – x} and B' = {u1, U2} for P1(IR), and suppose that 2 Q = ( -…

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Q: Verify by finding the matrix of S 0 T (a) by direct substitution and (b) by matrix multiplication of…

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Q: Find conditions on w, x, y, and z such that AB = BA for the matrices below.

A: Here we find the condition w,x,y,z such that AB=BA where A=wxyz , B=11-11

Q: Let A = [kaij]nxn, B = [aij]nxn, be an nxn matrices and k be a scalar then det(A) is equal to: *…

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Q: Let A and B benxn matrices such that AB = 1. Prove that |A| +0 and |B| + 0.

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Q: Let A E Mn(C) denote the n x n matrix with all entries equal to 1. Show that A is positive…

A: Let A belongs to  Mn (C) denote the n x n matrix with all entries equal to 1. Show that A is…

Q: Let A and B be two matrices of size 3 x3 such that det(A) = -2 and det(B) = 2. Then det(2A 2B") =

A: the solution is following.

Q: Prove: For all m,n E N, (A†)™ = (A")† holds for all m x n matrices A.

A: Given that m,n∈ℕ. Let A be an m×n matrix. Let A+=Z. We know that A+=Z is pseudo inverse of A if it…

Q: Let A and B be 4x4 matrices such that det(A) = 5 and det(B) = 3. Compute the determinant of…

A: Explanation of the answer is as follows

Q: Find the standard matrices for T T, o T and T' = T o T where T : R? → R°, T;(x, y) = (x, x+ y, Yy)…

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Q: Let A be an m x n matrix. Is the set K defined below convex ? Why or why not ? K = {y € R" : 3r E R"…

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Q: Find h and k, consistent: which makes matrix t. 6ud Sua = k |

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Q: A matrix Q ϵ Mnxn(R ) is called orthogonal if QQt = I . Prove that if Q is orthogonal, then det(Q) =…

A: Given data: The orthogonal matrix is given by "Q". The given condition is  QQT = I.....(1) Now, take…

Q: A be square matrix. I f A"= Onva for %3D n> 2. Then A= Onen

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Q: Let An be the n × n matrix given by If ∆n = det An, prove that ∆5 - 2 cos θ∆4 + ∆3 = 0

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Q: Assume that T and S are matrix of the same size. Prove or Disprove that (T + S) 2 is a symmetric,…

A: Given   T  and   S are matrix of same size. We have to prove if  T+S is symmetric, skew - symmetric…

Q: Is it true or false? |A+ B" |=| AT + B[T for any nxn matrices A and B. Birini seçin:

A: We are asked to find the statement is true or false.

Q: If the matrices A and B are conformable for sum A+B the product AB then prove that (A+B)" = A" +B"…

A: Prove that. (a).A+BT=AT+BT(b).ABT=BTAT