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Maths Help needed - Matrices

I'm not sure how to find r for this question. I have found the values of k for A, B and C (using the determinant as either = or =/= to 0). But I'm not sure how to apply my knowledge of about the geometric situations to find r.


Could someone help me with this?
(edited 1 year ago)
Theyd want you to think about each row representing a plane and what happens for the 3 planes (3 rows) for those different cases.

For r, you would look to see whether the equations were consistent or not. Rather than using tthe determinant, youre probably better off using the augmented marix for both k and r.
(edited 1 year ago)
Reply 2
Original post by mqb2766
Theyd want you to think about each row representing a plane and what happens for the 3 planes (3 rows) for those different cases.

For r, you would look to see whether the equations were consistent or not. Rather than using tthe determinant, youre probably better off using the augmented marix for both k and r.

I've never heard of the augmented matrix in further core pure 1, how would you use that to solve for r?
Original post by Ice.Log
I've never heard of the augmented matrix in further core pure 1, how would you use that to solve for r?

Where is the question from, what have you covered about showing a system of linear equations is consistent? Something like
https://byjus.com/maths/consistent-and-inconsistent-systems/
https://digestiblenotes.com/further_maths/matrices/systems.php

Using the determinant only tells you about the invertibiity of "A", you need to consider something like the same (row) operations on "b" (the right hand side) to work out whether the last row corresponds to something like 0 = 0.
(edited 1 year ago)
If you don't know what augmented matrices are this is probably the method best suited to you I think ?

If the determinant of a matrix in the form representing linear simultaneous equations is non-zero, the system of equations will have a unique solution, so for part a you need to find the determinant of the matrix and the value of k making it zero. The answer will be all real values of k not equal to this value. The value of r doesn't matter.

For b and c k must equal this value. You'll have 3 simultaneous equations which you can reduce to two equations with two terms. For infinite solutions they must be identical, giving you the value of r.

Geometrically, you should have learnt that for infinite solutions the three planes will either be identical or form a sheaf. For no solutions, two or more will be parallel, or otherwise the planes form a prism.
(edited 1 year ago)
Reply 5
Original post by mqb2766
Where is the question from, what have you covered about showing a system of linear equations is consistent? Something like
https://byjus.com/maths/consistent-and-inconsistent-systems/
https://digestiblenotes.com/further_maths/matrices/systems.php

Using the determinant only tells you about the invertibiity of "A", you need to consider something like the same (row) operations on "b" (the right hand side) to work out whether the last row corresponds to something like 0 = 0.

Thank you for the links, I've now learnt how to use the augmented matrix to solve this!
Reply 6
Original post by Driving_Mad
If you don't know what augmented matrices are this is probably the method best suited to you I think ?

If the determinant of a matrix in the form representing linear simultaneous equations is non-zero, the system of equations will have a unique solution, so for part a you need to find the determinant of the matrix and the value of k making it zero. The answer will be all real values of k not equal to this value. The value of r doesn't matter.

For b and c k must equal this value. You'll have 3 simultaneous equations which you can reduce to two equations with two terms. For infinite solutions they must be identical, giving you the value of r.

Geometrically, you should have learnt that for infinite solutions the three planes will either be identical or form a sheaf. For no solutions, two or more will be parallel, or otherwise the planes form a prism.

Thank you, I have used this method too to get k=/=7 for A, k=7 and r=7 for B, and then k=7 and r=/=7 for C.
And k and r both have to be real if they aren't 7.
Original post by Ice.Log
Thank you for the links, I've now learnt how to use the augmented matrix to solve this!


Good, but its worth chatting with your teacher about what you need to cover.
Original post by Ice.Log
Thank you, I have used this method too to get k=/=7 for A, k=7 and r=7 for B, and then k=7 and r=/=7 for C.
And k and r both have to be real if they aren't 7.

Agree about 7 being the important value for both k and r.
Reply 9
Original post by mqb2766
Good, but its worth chatting with your teacher about what you need to cover.

I looked through the Further Core Pure books (I do edexcel) and it doesn't appear in it. Our teacher has finished matrices now so I doubt I'll ever be taught about it unless I ask after school.
Original post by Ice.Log
I looked through the Further Core Pure books (I do edexcel) and it doesn't appear in it. Our teacher has finished matrices now so I doubt I'll ever be taught about it unless I ask after school.


Id ask about it. If the question is from the book, treating them as 3 simultaneous equations and reducing them to 2 as Driving Mad describes is the same as representing it as the augmented matrix and reducing the last row to zero , but its probably the "preferred" method. It would be worth being clear about.
(edited 1 year ago)
Original post by Ice.Log
I looked through the Further Core Pure books (I do edexcel) and it doesn't appear in it. Our teacher has finished matrices now so I doubt I'll ever be taught about it unless I ask after school.

Note that youve not uploaded your working, but probably the "easiest" way to do this one would be to note that if the 3rd row was a linear combination of the other two, so
row3 = a*row1 + b*row2
you'd want to find a and b and then use them to reason about k and r. Arguably the easiest way to find them is to use the first two columns so
|3 2||a| = |5|
|-1 1||b| |-3|
So invert the 2*2 matrix and multiply it by the [5 -3]' vector which gives the linear coeffs [a b]'. Then once you have a and b you can reason about the third column to get k and the vector on the right to get r. But check what you should do.
(edited 1 year ago)

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