linear algebra

Quiz 4

1) Use appropriate algorithmic way to justify if A is invertible. Find 𝑨−𝟏 if

that exists by using the algorithm. (6 points)

A = [

𝟏 𝟎 𝟎 𝟎 𝟐 𝟑 𝟒 𝟎 𝟐 𝟎 𝟐 𝟏 𝟏 𝟐 𝟑 𝟏

]

Now answer the followings with explanation:

(i) Is 𝑨𝑻 invertible? If so write it’s inverse without any elementary row

operation.

(ii) Let B and C be any two invertible 4x4 matrix. How many solutions

does the equation (𝑨𝑩𝑪)𝑻𝑿 = 𝟎 have?

2) Does the columns of the following matrix span 𝑹𝟔? [ DO NOT USE ROW

OPERATION]

[ 𝟏 𝟐 𝟑 −𝟏 −𝟐 𝟑 𝟏 𝟐 𝟑 −𝟏 −𝟐 𝟑 𝟏 𝟐 𝟑 −𝟏 −𝟐 𝟑 𝟏 𝟐 𝟑 −𝟏 −𝟐 𝟑 𝟏 𝟐 𝟑 −𝟏 −𝟐 𝟑 𝟏 𝟐 𝟑 −𝟏 −𝟐 𝟑]

(Hint: Use some definition and some theorem) (4 points)

3) Let A be the Matrix of the Linear Transformation T:𝑹𝟔 → 𝑹𝟔 given by

T ((𝒙𝟏, 𝒙𝟐, 𝒙𝟑, 𝒙𝟒, 𝒙𝟓, 𝒙𝟔)

= (𝟐𝒙𝟏 − 𝒙𝟐, 𝟑𝒙𝟏, 𝟐𝒙𝟑 − 𝒙𝟒, 𝟑𝒙𝟑, 𝟐𝒙𝟓 − 𝒙𝟔, 𝟑𝒙𝟔) . (5 points)

Is T invertible? If so find a formula for 𝑻−𝟏 as in the form of T. [ DO NOT

USE ROW OPERATION]

.