In this challenge, you will be given a square matrix A, a vector v, and a scalar λ. You will be required to determine if (λ, v) is an eigenpair corresponding to A; that is, whether or not Av = λv.

Dot Product

The dot product of two vectors is the sum of element-wise multiplication. For example, the dot product of the following two vectors is:

(1, 2, 3) * (4, 5, 6) = 1*4 + 2*5 + 3*6 = 32

Note that the dot product is only defined between two vectors of the same length.

Matrix-Vector Multiplication

A matrix is a 2D grid of values. An m x n matrix has m rows and n columns. We can imagine an m x n matrix as m vectors of length n (if we take the rows).

Matrix-Vector multiplication is defined between an m x n matrix and a size-n vector. If we multiply an m x n matrix and a size-n vector, we obtain a size-m vector. The i-th value in the result vector is the dot product of the i-th row of the matrix and the original vector.

Example

        1 2 3 4 5
Let A = 3 4 5 6 7
        5 6 7 8 9

        1
        3
Let v = 5
        7
        9

If we multiply the matrix and the vector Av = x, we get the following:

x₁ = A^T₁ * v /* AT1 means the first row of A; A1 would be the first column */ = (1,2,3,4,5) * (1,3,5,7,9) = 11 + 23 + 35 + 47 + 5*9 = 1+6+15+28+45 = 95

x₂ = A^T₂ * v = (3,4,5,6,7) * (1,3,5,7,9) = 31 + 43 + 55 + 67 + 7*9 = 3+12+25+42+63 = 145

x₃ = A^T₃ * v = (5,6,7,8,9) * (1,3,5,7,9) = 51 + 63 + 75 + 87 + 9*9 = 5+18+35+56+81 = 195

So, we get Av = x = (95, 145, 195).

Scalar Multiplication

Multiplication of a scalar (a single number) and a vector is simply element-wise multiplication. For example, 3 * (1, 2, 3) = (3, 6, 9). It's fairly straightforward.

Eigenvalues and Eigenvectors

Given the matrix A, we say that λ is an eigenvalue corresponding to v and v is an eigenvector corresponding to λ if and only if Av = λv. (Where Av is matrix-vector multiplication and λv is scalar multiplication).

(λ, v) is an eigenpair.

Challenge Specifications

Input

Input will consist of a matrix, a vector, and a scalar. These can be taken in any order in any reasonable format.

Output

Output will be a truthy/falsy value; truthy if and only if the scalar and the vector are an eigenpair with the matrix specified.

Rules

• Standard loopholes apply
• If a built-in for verifying an eigenpair exists in your language, you may not use it.
• You may assume that all numbers are integers

Test Cases

 MATRIX  VECTOR  EIGENVALUE
 2 -3 -1    3
 1 -2 -1    1    1    ->    TRUE
 1 -3  0    0

 2 -3 -1    1
 1 -2 -1    1    -2   ->    TRUE
 1 -3  0    1

 1  6  3    1
 0 -2  0    0    4    ->    TRUE
 3  6  1    1

 1  0 -1    2
-1  1  1    1    7    ->    FALSE
 1  0  0    0

-4 3    1    
 2 1    2    2    ->    TRUE

I will add a 4x4 later.

Unreadable Test Cases that are easier for testing