algebra-det.

a) Suppose that B is an n ¡Á n matrix, and k is a scalar. Give a formula for det (kB) in terms of det B .

b) Show that your formula from (a) is true for all n and for any k.

det (kB) = k^n det B

This is because the determinant is a multilinear function of it's columns (or rows) considered as vectors to the reals. If you multiply one particular colomn of the matrix by k, then the entire determinat get's multiplied by k.

So, if you muliply the whole matrix by k, then that amounts to multiplying each column by k, and thus the determinant picks up a factor k^n.

  1. 👍
  2. 👎
  3. 👁

Respond to this Question

First Name

Your Response

Similar Questions

  1. Linear algebra

    please help me with this question I have no idea how to solve it, thanks Suppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is rotation by 1/6π.

  2. math

    verify The set of all 2 × 2 invertible matrices with the standard matrix addition and scalar multiplication is a vector space or not?

  3. Algebra

    Given the following vector X, find a non-zero square matrix A such that AX=0: You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. X= 2 -8 6 A= _ _ _ _ _ _ _ _ _ Please help, I

  4. Math

    Suppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is reflection over the line y=−3/2x. Please give me the solution

  1. physics: scalars/vectors

    Please check. 1. Which of the following quantities are scalars, and which are vectors? a. the acceleration of a plane as it takes off- VECTOR b. the number of passengers on the plane- SCALAR c. the duration of the flight- SCALAR

  2. Math

    Given the following matrix A, find an invertible matrix U so that UA is equal to the reduced row-echelon form of A: You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. A = 3 3

  3. linear algebra

    3. Suppose A is symmetric positive definite and Q is an orthogonal matrix (square with orthonormal columns). True or false (with a reason or counterexample)? a) (Q^(T))AQ is a diagonal matrix b) (Q^(T))AQ is a symmetric positive

  4. Augmented Matrix

    Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent, dependent, or inconsistent. DO ALL WORK BY HAND. x + 2y + 4z = 6 y + z = 1 x + 3y + 5z =10 If one subtracts the

  1. Diagonalize

    construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible. Just write down a diagonal matrix with one zero on the diagonal and then apply an orthogonal transformation. E.g. if you start with the matrix: A = [1

  2. Math

    Multiplying a vector by a scalar results in: a) a scalar b) a perpendicular vector c) a collinear vector c) a parallel scalar

  3. Math

    Directions: Use the following matrix to perform the elementary row operations sequentially. A=[3 2 |8] [5 2 |12] 1.) (1/3) R1 From the original matrix 2.) -5R1+R R2 From matrix in question 1.

  4. statics/physics

    Figure: session.masteringengineering.com/problemAsset/2086595/7/05Ch_Hibbeler_StaticISM_4th_11.jpg Consider the beam shown in (Figure 1). Suppose that w = 390 N/m . Determine the x and y components of reaction at B using scalar

You can view more similar questions or ask a new question.