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Which of the following best describes the two statements about a 6 x matrix whose columnsare linearly independent:A) The nullity of A is 0 B) The rank of A is 4Only...

Question

Which of the following best describes the two statements about a 6 x matrix whose columnsare linearly independent:A) The nullity of A is 0 B) The rank of A is 4Only A is trueOnly ' B is trueBoth A and are trueBoth and are falseConsider the set e defined as S =d-%k and addition on this set defined by ax [Je[;]-[ by and scalar multiplication defined by ~[]-[6]Which of the following best characterizes the two statements:The set $ is closed under addition The set $ has an additive identityOnly

Which of the following best describes the two statements about a 6 x matrix whose columns are linearly independent: A) The nullity of A is 0 B) The rank of A is 4 Only A is true Only ' B is true Both A and are true Both and are false Consider the set e defined as S = d-%k and addition on this set defined by ax [Je[;]-[ by and scalar multiplication defined by ~[]-[6] Which of the following best characterizes the two statements: The set $ is closed under addition The set $ has an additive identity Only Ais true Only B is true Both and are true Both A and B are false



Answers

a. Fill in the blank in the following statement: "If $A$ is an $m \times n$ matrix, then the columns of $A$ are linearly independent if and only if $A$ has ______ pivot columns.”
b. Explain why the statement in (a) is true.

A square minus B square is a minus B into A plus B. Yeah. Opening the bracket, we have a square minus B plus A B minus beasts, players, I'm sorry, so this is equal to a squared minus B square implies a B equal to be for option B. S. Sculpt. Thank you.

So the easiest way to answer this question is to simply look at matrix multiplication and the way it works, and then we'll understand how our rows and columns need to be set up. So if we're multiplying metrics that attempts to matrix being we would start with the first row of a in the first column of B who would multiply through first, this one, then this one in this three times, this one, this one in this one that would give us our first position. Then you would move to the next column and Billy staying the same row in a so that will fill our next position. Then we would move to the next row in a and then we repeat through the columns and be so since we're looking, we go row by row and a and we're going column by column and be We know that in order to do matrix multiplication, we need to have the same number of rows. So number of rose in a as we have columns and be

So the easiest way to solve this question is to work with Cem Matrix multiplication. So we'll call the first Matrix a 2nd 1 b. So a Times B is going to yield us a single number of matrix. And that's because we only have one Rohan A and one column and be now to solve this. We would look through all of the first row of a then we go through all of the first column of B. I only have one of each, but basically we're just do one times one plus two times two plus three times three, which would give us 14. This would be our answer, but this No looking at the question that we're actually trying to solve, you see that A has three columns and be his three rows. So that's the necessary requirement. We need to have the same number of so number of columns and the first matrix a must be equal to in number of Rose in our second Matrix B

A and two B. R. Two non singular mattresses. Non singular matrices of the same model such that areas to end equal to I. Then for positive injuring greater than one be raised to a sorry B. A restaurant minus one into being worse minus B. A inverse being worse. We need to find the value of this. Yeah. Since there is to an equal to I. Which implies and you arrest you and -1 equal to I. We have a race two in -1 equals two. A inverse product of two matrices equal to an ID intimated symbolize then was to each other no be arrest you and minus one B in both minus B. A. And was being was equal to. We know that a restaurant -1 is equal to him. Was substitute here be in verse being worth minus B. In this being was That is zero the financing option.


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