GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 20

Answer 1

$ABCx = b$ has solution $x = \begin{bmatrix} 7\\1 \\2 \end{bmatrix} +\alpha \begin{bmatrix} 2\\3 \\-4 \end{bmatrix}$

So,$ ABCx^{‘} = 0 $

where $x^{‘} = \alpha \begin{bmatrix} 2\\3 \\-4 \end{bmatrix} $ are all the solutions for ABCx= 0.

$A^{-1}ABCx^{‘} = A^{-1}0$ (as A is invertible i.e $|A| \neq 0$)

$BCx^{‘} = 0$
$By = 0$

where y = Cx’ = $\begin{bmatrix} 4 &1 &1 \\ 0& 1 & 1\\ 0 & 0& 2 \end{bmatrix} * \begin{bmatrix} 2\alpha \\ 3\alpha \\ -4\alpha \end{bmatrix} = \alpha \begin{bmatrix} 7\\-1 \\ -8 \end{bmatrix}$ ,

So, B has nullity = 1 (i.e. one free column)

Also B is a 5x3 matrix (calculated from matrix multiplication rule)

We know : Rank + null space  = number of columns

Rank = 3 – 1 = 2

$Rank (B) = 2$

Comments

How ABCx' = 0?

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How ABC x' = 0 ? And why x' = alpha * [2

                                                                      3

                                                                     -4]

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Theorem. Let A be an mxn matrix, and b an m×1 vector. If y0 is a solution to Ax=b, then every solution of Ax=b is of the form y0+s, where s is a solution to Ax=0, and every such vector is a solution to Ax=b. In other words, the complete list of solutions to Ax=b is given by finding a particular solution y0 to Ax=b, and adding the solutions to Ax=0.

source : https://math.stackexchange.com/questions/4020850/relation-between-ax-0-and-ax-b

Answer 2

Given Matrix A is 5*5 and Matrix C is 3*3. Hence Matrix B will be 5*3.

we know,

Rank of Matrix K of order m*n = min(m,n).

so, max possible Rank of Matrix B 5*3 will be 3.

But in the question it is given x is having some free variable which is α1.

we know,

Rank =Total no. of colums – Free variable (L.I. vector).

Rank of B is 3 – 1 = 2

Ans is 2.

Comments

We can’t directly say anything about null space of B ,

You got the correct answer because C was invertible what is that was not the case

(i.e.what if C itself has some LD columns ) so your approch to this question is not correct

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I guess you are getting answer correctly in this case but free variable doesnt compulsorily directly imply about LD rows in B