Express the linear function as F(x) = CX for some matrix C. f(x1,x2,x3,x4) = (5x3 - x1, 2x3 + x4, -3x2 + x1, x1+ x2 - x4).

Q: Prove that det : M2×2(F) →F is a 2-linear function of the columns of a matrix.

A: It is required to prove thatDet: M2×2(F) -> F is a 2- linear function of the columns of a…

Q: Let Q : R2 R be the quadratic form defined by Q(x) = 6x² – 12x12+3. 7. %3D (a) Write down the…

A: Quadratic form problem....

Q: Find x such that the matrix is singular. A = -3 -2 X =

A: Given A = 6x-3-2

Q: Give a matrix representation of the quadratic form f(x, y, z) = 4x 2 + 9y2+ 16z 2 − 2xy + 4xz − 6yz…

A: Given: f(x,y,z)=4x2+9y2+16z2-2xy+4xz-6yz             We have to find a matrix representation of the…

Q: Find the symmetric matrix A associated with the given quadratic form.…

A: Solution: Consider the given quadratic form is: 2x2 - 3y2 + z2 - 4xz we know that if ϕ =…

Q: Find the matrix M of the linear transformation T : R³ → R² given by -2x1 – x2 + 7x3 T X2 2x2 + 3x3 M…

A:

Q: Find x such that the matrix A is equal to its own inverse.

A: We have A=-2-33x Now, A-1=1AadjA For a 2×2 matrix A=abcd, we have A=ad-bc  , adjA=d-b-ca Hence, for…

Q: Are these matrices linearly independent? If not then give a dependence relation. 1 2 2 3

A:

Q: Use the determinant of the coefficient matrix to determine whether the system of linear equations…

A: Given system of equations is We know that if determinant of coefficient matrix is nonzero, then the…

Q: Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists)

A: Given matrix is,

Q: Find x such that the matrix is singular. X 3 A = -7 2 X =

A: Singular Matrix - A matrix is said to be singular if and only if its determinant is equal to zero.

Q: Find the values of for which :=,0) and": =(a.1) are 4a-4 linearly independent in *".

A: To find the values of a for which the below vectors are linearly independent in R2. v1=4a−4, av2=a,…

Q: Find three non-trivial 2 x 2 matrices such that the square of the matrix is the zero matrix, i.e: A?

A: We want to find matrix A such that A2 = 0000

Q: Find the matrix of the quadratic form. Assume x is in R2. a. 1x21 – 4x1x2 + 5x22 b. 3x21 + 2x1x2

A:

Q: Solve for the value of x from the given matrix: [(x - 2) y (2z − j)] = [(y+z) j2 (x-1)]

A:

Q: Find the B-matrix for the transformation x Ax, when B = {b1, b2}. [ -1 4 3 b2 = A = b1 = -2 3

A: Given matrix, A=-14-23B=32-11

Q: Q1): Prove that: A square matrix B has inverse if and only if determinant of B is not equal zero.

A:

Q: Find all pairs (x,y) for which matrix B is not invertible, where 1 1 B = 2 1 1 0 -1 y

A: Here we will apply the condition for a matrix not being invertible to solve this question that is…

Q: Find x such that the matrix is singular. [ х 7 A = -5 2 X =

A:

Q: Find x such that the matrix is singular. [ х 3 A = -9 8 X =

A: The given matrix is: A=x3-98

Q: Find all stationary points of the function f(x,y)= 3x2 -6xy+2y3, and use Hessian Matrix and the…

A: Given function,

Q: Find the standard matrix of the linear transformation. -y T y х+ 5у Зх — 9у

A: GIvenTxy=-yx+5y3x-9yTo find Matrix of linear transformation.

Q: For the quadratic form x²+ xz - 6yz = 0 in the matrix notation X'AX the matrix A is: 1 1

A:

Q: Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists).

A: Given:0111 over ℤ2

Q: Find the standard matrix A of linear transformation T:R3 → R? defined by, [X1 – 2x2] [2x1 + x3] TX2

A:

Q: (E) - 4x – Y M [ y + x

A: The solution is given as

Q: Find x such that the matrix is singular. A = -3 -4 X =

A: Matrix is said to be singular , if and only if the matrix has determinant is zero Det(A) = 0

Q: The matrix X1 X = is called a vector.

A: We have to find

Q: Find the matrix M of the linear transformation T : R² → R² given by -8x1 + 9x2 5x1 – x2 x1 X2 M =

A:

Q: Q1): Prove that: A square matrix B has inverse if and only determinant of B is not equal zero.

A:

Q: Determine the values of x of a given linear system shown below using cramer's rule. X1 - X2 - X3 +…

A: We will find out the required solution.

Q: Solve for the value of x from the given matrix: [(x-2) y (2z - j)] = [(y+z) j2 (x-1)]

A:

Q: (b) Find the Jacobian matrix ду əx where y [1+1+2+3 -x1x2x3 -[1+5+ = -B X2 I3 and x =

A: (b) Given y=1+x1+x2+x3-x1 x2 x3 and x=x1x2x3   To Find: The Jacobian Matrix ∂y∂x

Q: Find the symmetric matrix A associated with the given quadratic form. x1x2

A:

Q: Reduce the quadratic form x2+y²-9xy into a canonical form of the matrix.

A: The given equation is x2+y2-9xy. The standard equation is ax2+by2+cz2+2fyz+2gzx+2hxy. The…

Q: For a nonsingular matrix A and nonzero scalar k, show that (k A)−1 = 1 k A−1.

A: Given that A is nonsingular matrix. That is, A≠0. Since k is a non zero scalar, kA≠0. Therefore, kA…

Q: find if the vector x is in the range for matrix A

A:

Q: For each matrix A with real valued entries, find a diagonal matrix D and an invertible matrix Q such…

A: Part(a): Given matrix is, A=1332 Now, finding the eigenvalues of the given matrix,…

Q: hat is the matrix for

A: Given, consider the two-tank system shown below. Each of the two tanks contains 200 gallons of water…

Q: Let A be the matrix 8 Find p(A). p(x) = x – 2x + 4 i p(A) =

A:

Q: Let A = (a;;)ij be an m x n matrix of real numbers, and let f : R" Rm be defined by 0-0 x2 x2 f = A

A: To find the derivative of f given: matrix A which has m rows and n columns  also the function f…

Q: Use the Finite Difference method to write the equation x" + 2x² - 6x = 2, with the boundary…

A: Given the equation  x''+2x'-6x=2x0=0, x9=0

Q: Find a 3x1 matrix X with entries not all zero such that AX = X , where [1 0 -1] A =1 0 1 4 -4

A:

Q: Find the symmetric matrix A associated with the given quadratic form.

A: To determine: The symmetric matrix A associated with the given quadratic form. 5 x12 - x22  + 2 x32…

Q: Determine the standard matrix of a reflection in R2 in the line -x1 + 2x2 = 0.

A: Given line is -x1+2x2=0

Q: Consider the linear function T below. 4 T: R* → R*, X1 x1 + V5x2 X2 X2 X3 х2 — 5хз X4 -V5x4 (a)…

A: