-4 Let A = Find a matrix B such that AB=I. 1 - 1 1 O a) 4 O b) 1 1 4 3. c) - 3. 3 1 4 3. o d) 3. 4 3. 3

Q: Prove that if B is the matrix obtained by interchanging the rows of a 2 ×2 matrix A, then det(B) =…

A: Let the matrix A and the new matrix B be as follows.

Q: a Verify that det EA = (det E)(det A), where E is the elementary matrix shown and A =

A: E=1k01  A=abcd

Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is as follows

Q: B) Suppose A is an invertible matrix inM, (R). And x, y E R". Prove it: det(A+ ry") = (1+y"…

A: Given: A is an invertible matrix in Mn(ℝ) x,y∈ℝn To prove: det(A+xyT)=(I+yTA-1x)det(A)

Q: 1) Construct the matrix A =(a;;) where 3x2 ) aj = i+j, if i = j,and Li-j, if itj %3D

A:

Q: Let E and F be n × n elementary matrices and let C = EF. Is C nonsingular? Explain.

A: Let E and F are elementary matrices and let

Q: Let A be an n xn matrix, then for any r ER, det(rA)= 1 det A.

A:

Q: Let A = - 2     1    0                 1     0    1                2    -3    1        A is a…

A: trace is the sum of all diagonal elements of the matrix.

Q: Let A,B be 3*3 matrix such that, det(A)=4, det(B) =5, find det(2AB)= 100 Select one: O True False

A: Given  det(A) = 4 det(B) = 5

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".

A:

Q: 3. (a) Let In be the n x n identity matrix. What is det(-2In)? (b) If A is an n x n matrix and c ER…

A:

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: Prove the formula for a nonsingular n × n matrix A. Assume n ≥ 2.∣adj(A)∣ = ∣A∣n−1

A:

Q: Let A,B be 3*3 matrix such that , det(A)=4, det(B) =5, find det(2AB)= 100 Select one: O True False

A:

Q: Let A be an nxn matrix. Prove that dim(col(A²)) < dim(col(A)).

A: Given: Assume that, A is an n×n matrix.

Q: b) Prove: Let A be an m × n matrix, R" = NS(A) RS(A).

A: Given: Let A be an m×n matrix , Rn=NSA⊕RSA

Q: 11. Let A be an m x n matrix and let x, y e R". Prove that if Ax = 0 and y = ATb for some b e R",…

A:

Q: b) Prove: Let A be an m x n matrix, R" = NS(A) + RS(A). %3D

A: A be an m×n matrix, we have to prove that ℝn=NSA⊕RSA  …1.

Q: . Prove that (i) . =1, and (ii) ||, =\n (I is the Identity Matrix of order n).

A:

Q: **32. Suppose A is an m x n matrix and x e R" satisfies (ATA)x = 0. Prove that Ax = 0. (Hint: What…

A: Here we use property of modular and dot product.

Q: 1 Let A be a nonsingular 4 x 4 matrix such that |A-| = -2, then 4 Select one: O True O False

A:

Q: Let A be an nxn matrix. If A-= det A adj A is computed, what should AA be equal to in order to…

A: Using the given matrix equation we need to find the value of AA-1 .

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

A:

Q: I find the cooridinate matrix for A realtive to S=SA, Ag „Az, Au? where A= [2 0 and 1 1 00

A: The coordinates matrix for A is [a b c d]=aA1+bA2+cA3+dA4=A .see 2nd step.

Q: 2. Verify, without using the matrix for 0(), that [O()]° is the identity matrix 1 0 0 1

A: This is a problem of rotation of the matrix.

Q: snip

A:

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

A:

Q: Let A be a 2 x 2 matrix. If det (A) # 0, then det (adj A) = det (A).

A:

Q: Let A be the 2 x 3 matrix with entries aij = (i – 1)(j+ 1). Which of the following is the matrix A?

A: A be a 2×3 matrix. This means A2×3 . entries of matrix A are ai,j=(i-1)(j+1)

Q: Let A be an n × n matrix. Show that if Ak+1 = O, then I − A is nonsingular and (I − A)−1 = I + A +…

A:

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

A:

Q: 8. Prove that (i) ||7, =1, and (ii) |7I, =\n (I is the Identity Matrix of order n).

A:

Q: Let A be an invertible diagonal matrix. Which of the following is false: O (2A)T = 2 AT If A-1= A,…

A:

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: Prove that (i) 7|, =1, and (ii) ||, =\n (I is the Identity Matrix of order n).

A:

Q: Let U be a square matrix such that u'U=I. Show that det U = +1. Assume that U'u=1. Since the desired…

A:

Q: Which of the following is the LU-factorization of the matrix 1 0 a) Бая 0 1 0 -1] 1 1 0 b) :] 01 0 1…

A:

Q: 16. Prove that for any square matrix A, ||A||ž E ||A* A||2.

A: Given: A is a square matrix.

Q: Determine the matrix C such that (CT – 21) 1 1 2 2 3

A:

Q: Find the value of k such that matrix A is 2. k – 10 singular on Ik - 5 а. 1 С. 3 b. 2 d. 4 O D O B O…

A: Topic = matrices Correct option=D

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

A: Suppose A be any square matrix of order n such that An=On×n     for n&gt;2

Q: Let A be an nxn matrix, then for any r ER, det(rA)= –det A. Select one: O True O False

A:

Q: Let A = 2 O 5 31 O-36 5 Find unitary matrix U and a diagonal matix D Such that U₁AU = D.

A:

Q: Let A, Band C be square matrices Such that AB = AC, If A#0,then B-C. 1s this True or false? Explain…

A: Let A, B, and C be square matrices such that AB=AC. If A≠0 , then B=C. Determine true or false.

Q: a b Verify that det EA = (det E)(det A), where E is the elementary matrix shown and A =

A:

Q: 21 1 2 1 ol -1 A = | 0 2.

A:

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

A: