1). Determine whether the matrix operator T: R → R³ defined by the equations is One-One and Onto. If so, find the standard matrix for the inverse operator and find T-(w1, w2, w3). wi xi – 3x2 + 43 W2 -x1 + x2 + x3 W3 -2x2 + 5x3 ||||

Q: Let Cand D be two n Xn invertible matrices. Find the xN matrix X in terms of Cand D such that…

A: A matrix is said to be invertible if inverse of matrix exists. Multiplication of matrix with its…

Q: The matrix of cofactors of A is precisely [adj(A)]' O True O False

A:

Q: Let f1, f2, f3 be functions in F(R). a. For a set of real numbers x1, X2, X3, let (f(x,)) be the…

A: Given that, f1,f2,f3  be the functions in F(R) It is given that fi(xj) be the 3 times 3 matrix.…

Q: The linear operator T : R3 → R3 is defined by T(x1, x2, x3) = (w1, w2, w3), where w1 = 2x1 + 4x2 +…

A: Given  Three equations in 3 variables. W1 = 2x + 4y + z W2 = 9y + 2z W3 = 2x - 8y - 2z

Q: Determine whether V and W are isomorphic. If they are, give an explicit isomorphism T: V--->W. V…

A:

Q: CD, where C and D are n x n matrices and C is invertible. Prove that the matrix product DC is…

A:

Q: How is the matrix of R^−1 obtained from that of R in general, that is, when A = {a1, . . . , am}, B…

A:

Q: 2. Prove that a 2 x 2 matrix A with entries in Z26 is invertible if and only if gcd(det(A), 26) = 1.

A:

Q: A and B are n X n matrices If Bis invertible, prove that det(B-1AB) = det(A)

A: A and B are n X n matrices and B is invertible.

Q: 3. (a) If A be an n x n invertible matrix. Prove that the inverse of A is unique.

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Let [1 0.25] A : 1 Find an invertible matrix X and a diagonal matrix D such that X-'AX = D.

A: If for matrix A,      X-1 AX =D  Then matrix A is Diagonalisable,   And where D is diagonal matrix…

Q: Prove that a 2 X 2 matrix A is reducible. if and only if a12=0 and a21=0

A:

Q: Let A be a nonsingular matrix. Show that det(A−1) = 1/det(A)

A:

Q: If A is an invertible matrix. Then det(AA-): 1D

A: IA is an invertible matrix To find det(AA-1)

Q: Give an example of an operator whose matrix with respect to some basis contains only O's on the…

A:

Q: 1. Let T,(x1,X2,X3) = (4x1, –2x1 + X2, -X1 – 3x2) and T2(x1,X2,X3) = (x1 + 2x2, -X3, 4X1 – x3). a.…

A:

Q: Let A be a 3 x 3 matrix such that det(A) = 2. Evaluate det(Aª) and det(2A).

A: Recall the following. If A is a n×n matrix and m be any positive scalardet k A=kndet Adet Am=det Am

Q: Let A be an n×n invertible matrix. Show that det(A-1)=(detA)-1

A: Given  A be an n×n invertible matrix i.e.  To show

Q: [4 Ja - Determine whether the linear operator T:R² → R² defined by the equations is One - to - one ,…

A: a) The linear operator  T : ℝ2→ ℝ2 defined by the equations…

Q: The matrix that implements the mapping T (₁, 2) = (-3x1 +4x2,0, 7x2 - 5x1,1 + 2x₂) is -3071 4 052 0…

A:

Q: Show that ∣A∣ = ∣AT∣ for the matrix below.

A: Now the transpose is,

Q: Let A be a 3 × 3 matrix such that Atr A = 4I3. Prove that det A = ±8. (Here, I3 is the 3 x 3…

A:

Q: If A is a 2×2 matrix such that trace(A) = det(A) = 3, then %3D value of trace of (A-!) ?

A: Introduction: It is the sum of the diagonal elements of the matrix from upper left to lower right.…

Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: If A be an n x n invertible matrix. Prove that the inverse of A is unique.

A: The assumption for the matrices helps to prove the required. A matrix is said to be invertible if…

Q: Prove that if A is an invertible matrix then det(A−1) = 1/det(A)

A:

Q: Let M₂ (R) be the set of 2 x 2 matrices with real number entries, i.e., b M₂ (R) = { [ a &] |…

A: The given matrix is M2ℝ=abcd: a,b,c,d∈ℝ To find : (a) M2ℝ is a ring under addition and…

Q: Determine whether the given linear operator/matrix is diagonalizable. (a) T: R³ → R³ with T(1, 1, 1)…

A:

Q: Show that if A is an orthogonal matrix, then det(A) = +1.

A: To prove, detA=±1

Q: 2. Determine whether the matrix operator T: R2R2 defined by the equations is invertible; if so, find…

A:

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: Let A be a nonsingular matrix. If det(A)=3, then det(A) = %3D

A: In this question we have to find detA-1.

Q: Let A = {a, b, c, d} and let R and S be the relations on A whose matrices are (1 0 1 0) 1 0 0 1 1 0…

A: The set is A = {a, b, c, d}. The matrices representing the relations R and S, respectively, are as…

Q: Let A be a nonsingular k x k matrix with k> 1. Show that. det [adj(A)] = [det(A)]*-1

A:

Q: Let Mı5(IR) = {A/A and 15th order square matrix with real elements} and consider the linear…

A:

Q: 3. Prove that (i) /, =1, and (ii) ||, =Vn (I is the Identity Matrix of order n).

A:

Q: (4) Let A be an invertible matrix. Prove that if A = Q¸R1 decompositions of A and if the diagonal…

A:

Q: Let T1: R$ -> R$ and T2: R5 -> R3. Give an explicit example (show me the matrices!!!) of T1 and T2…

A:

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

A: Given: A is a square matrix.

Q: 6. Let T: P₂ P₂ be the linear operator defined as T (p(x)) = p(3x), and let B = {1, x, x²} be the…

A:

Q: The matrix of cofactors of A is precisely [adj (A)]T. O True False

A: I have given the answer in the next step. Hope you understand that

Q: 6. Show that (AA)* = X* A* for all A E C and all matrices A.

A:

Q: Please see image

A: Given matrices are

Q: Let A be a nilpotent matrix (that is, Am = 0 for some m > 1). Prove that if A is diagonalizable,…

A: Consider a nilpotent matrix A. So Am=0 for some m>1.

Q: Let H: R* → R2 be defined by H(x, x2, X3, X4) = (x3 + x4, X1). Find the standard matrix for H. (H is…

A:

Q: show that if At exists for matrix A > then A is to

A:

Q: Determine whether the given linear operator/matrix is diagonalizable. (a) T : R³ → R³ with T(1, 1,…

A:

Q: 1. Let f(r) = ar +bx² + cz+d be a polynomial such that f(1) = 0, f(-1) = 12, f(2) = 12 and f(-2) =…

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.