Further Maths Matrices Question

Specifically to your question:
So hopefully you know this system can be rewritten into the form Ax=b (do you know what this means?)

If det(A) is not zero, A is invertible, and the solution vector is precisely x=A^{-1}b. Which means we have a unique solution.
So for the system not have a unique solution, det(A) = 0.

Here's a key point: det(A) = 0 only guarantees no unique solutions. Whether the system has infinitely many/no solution, we don't know.

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Now I actually don't think finding det(A) is the most efficient way. I'd straight-up do Gaussian elimination... but it looks like you haven't learned it yet?

Thank you for the help! The step I was missing was what you said; because there are infinitely many solutions, the determinant is therefore 0. I don't know why that information completely left my brain when i did this question.

I've been using fmsp's videos to study the content, and Gaussian eliminations haven't come up in those (or the textbook I think). It's interesting to see that other methods exist for a problem like this.

I was doing this question and couldn't figure out how to do it, so I looked at the SolutionBank answer online and for some reason they set the determinant to zero?

(q13 from mixed exercise 6 of the edexcel core pure textbook)

When I attempted it, I found the determinant (5a+15) as part of an attempt to find the inverse matrix, but I don't understand why setting it to equal 0 is what you need to do here?

solution bank answer:
https://www.activeteachonline.com/default/player/document/id/721178/external/0/uid/357726

Any help (or any advice for studying / revising FM in general) is great

We want to find the values of the real constants "a" and "b" for which the system has infinitely many solutions. This happens when the three equations are dependent, meaning one of them can be expressed as a linear combination of the other two.

To check if the equations are dependent, we'll use the method of determinants. The determinant of the coefficients of the variables is called the determinant of the system. If the determinant is zero, then the system has infinitely many solutions.

The determinants are as follows:

Determinant of the coefficients:
D = | 2 3 1 |
|-1 1 2 |
| a 1 4 |

Now, to find the value of "a" and "b" for which the system has infinitely many solutions, we need to set the determinant D to zero:

D = 0

Expanding the determinant:

D = 2(1 * 4 - 2 * 1) - 3(-1 * 4 - 2 * a) + 1(-(-1 * 1 - a * 2))

D = 8 - 3(-4 - 2a) + (1 - 2a)

Its worth having the read of the posting guidelines sticky at the top of the forum, especially not posting solutions. Youve gone wrong in the initial determinant calculation.

That looks like a ChatGPT response...

That looks like a ChatGPT response...

That looks like a ChatGPT response...