1. D [-1 0 1] B 2 1 5 9 4 a) Use the matrices A = |2 0 3 0-21 8 -2 to find the values where possible: al) AT+ B a4) АВ - С ti a2) A - C a5) BDT a3) AD аб) В'В

Q: a) For invertible matrices A, BER"Xn, show that ((AB)")- = (A-)"(B)".

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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: Suppose that A (A, then det(B)- Select one: A. 15 OB. 5. OC -15 ODO th ) is a (4 x 4) matrix, where…

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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: the matrIX A into a product QDQ" where Q is orthogonal and D is diagonal.

A: Given,          A=-5-1-1-1-3-1-1-1-3

Q: Suppose that Q is an orthogonal matrix, then O a lQ°| = 4 O b. Q°| = 2 O d. Q°| = 0

A: This question is related to Linear Algebra

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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: 4. Prove: If b is a non-zero vector, and x and x are both solutions to the matrix equation AX =6,…

A:

Q: Let A be an (n x n) matrix and za column vector. If AA = I then Ax|| = ||r||. %3D %3D O True O False

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Q: 4. Verify whether the two given matrices are similar. Justify your an- Swers Question a. 1 A = 1 and…

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Q: State Cayley-Hamilton theorem and prove this for following matrix [1 1 A 1 2 -1 %3D [5 8

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Q: Use A-1 =(1/detA)adj A to compute the in­verse of the coefficient matrix x + y - z = 1      x+y+z=2…

A: The coefficient matrix is, Now,

Q: Consider the matrices A = [₁, B = 3₁ C= ¹), D=1 K₁ = [5], K₂=[₁²2] Solve AX + BY = K₁ CX + DY = K₂…

A: Concept: We have the matrices A=3211 , B=0110 , C=111-1 , D=11-14 Now the matrices X and Y are…

Q: Show that if A is an n x n matrix, then A" can be written as a linear combination of the matrices I,…

A: Cayley-Hamilton Theorem:- Every square matrix satisfies its own characteristic equation.

Q: Let B = {b1, b2} and C = {c1, c2} be two bases for R², where [1 C2 7 3 3 b1 b2 3 10 22 b7 what is…

A:

Q: (b) Explain why in general, if u = [u uz] is a row of length 2, A = a1,1 a1,2 [@2,1 @2,2] we always…

A: Answer

Q: Use A-1 =(1/detA)adj A to compute the in­verse of the coefficient matrix x + y = 1   x-y=2

A: x+y=1 x-y=2 We need to find the inverse using A-1 =(1/detA)adj A

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: If both matrices A,B are orthogonal prove that (ab) is also orthogonal

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Q: If A and B are n x n matrices, and vector v is in the image of both A and B, then i must be in the…

A: True or false.

Q: 1% B = [ c = |; 1= 6 1, where j = v=1, express (a) Q3/B: If A = A.B, (b) B. C, (c) C.A and (d) A² in…

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Q: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

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Q: Find a matrix B such that AB = I, where [ ] 8 1 1 0 A = and I = -8 8 0 1 1 a b Hint: Let B = %3D c d…

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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: 2rị + 13 T: R' → R?, T 21, 22, 13, L4 ER -12 - 3r4

A: Given  and the standard bases are given by

Q: Let B = {b₁,b₂} and C = {c₁, c₂} be bases of R2. Find a change of coordinate matrix from C to B,…

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Q: Find two non zero matrices A and B such that AB = 0. %3|

A: Find the two matrix which satisfy the given condition

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: Suppose AB = C, where A and C are upper triangular. (a) Find 2 × 2 nonzero matrices A, B, C, where B…

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Q: 1. For any 3x3 matrices A & B: Verify that (AB)" = B"A". %3D

A: By using the properties

Q: Prove that if A is row-equivalent to B and B is row-equivalent to C, then A is row-equivalent to…

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Q: 2- Consider the matrices A = [₁²1], B = [₁ J₁ C=__=₁} K₁ = [50], K₂ = [₁²2] Solve AX + BY = K₁ CX +…

A: Given: The matrices are A=3211 ,B=0110,C=111-1, D=11-14,K1=5000 and K2=21102. To find: The matrixes…

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Q: Let A and B be n x n matrices and c be a scalar. Prove that, if B is obtained by multiplying a row…

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Q: Generalize the result by proving that, for any comfortable non-singular matrices A,B and C the…

A: Consider the given information. ABC-1=C-1B-1A-1 Multiply by pre and post C.…

Q: Q3, Write any matrices A and B of order 3 x 3 and verify the results Let A and B be n x n matrices,…

A: Write any matrices 'A' and 'B' of order 13×3 A=110012003    B=013100100 To verify 1)…

Q: Find two nonzero matrices A and B such that AB= 0

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Q: [1 0 2 31 l4 5 B = 6 ol,c = l6 ál, and V = |1|. What is the sum of the entries of the Lo. matrix VT…

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Q: Consider the matrix B given by 0 0 0 B = 1 0 0 0 1 0 Show that B3 = 0. Also, compute for the product…

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Q: 4. Verify whether the two given matrices are similar. Justify your an- Swers Question a. 1 1 3 and B…

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Q: Find all a and B >0 so that the matrix B = 23 6 2 is diagonally dominant;

A: A square matrix is diagonally dominant if   for .

Q: Let A be nxn orthogonal matrix and v E R". Then ||Av||? = || ||?. %3D Select one: True O False

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Q: 2. Compute the matrices D, P such that D =P-'AP. Verify the equation holds. [7 5 0 A 1 0 1

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Q: Find two unequal 2 × 2 matrices A and B such that

A: The objective is to find two different 2 × 2 matrices A and B such that AB=1001 Let A=pqrs, B=tuvw…

Q: Let M,xn be a set of square n x n-matrices. Is the set M„xn a vector Explain. Find 2 different bases…

A: We have to solve given problem:

Q: Show that for matrices A, B E RXn, a vector w = ABx can be written as w = E,(b{ x)a;, %3D :D1 where…

A: Since, A,B∈ℝn×n. ∴ A, B have 'n' rows and 'n' columns with entries as real numbers. The given vector…

Q: Suppose that a 3 x 3 matrix A factors into a product: 1 0 0111 U12 U13] A = 121 1 0 0 U22 U23 Ll31…

A: Given that  A = 100l2110l31l321 u11u12u130u22u2300u33 We have to determine the value of  det A.

Q: Find all a and B> 0 so that the matrix 4 a 28 5 4 2 is strictly diagonally dominant.

A: In a diagonally dominant square matrix, the magnitude of the element corresponding to the diagonal,…