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1. Hi, the question asks to find the determinant for this 4 by 4 matrix. I think I have used the long method to calculate this by focusing on the first column which is the only column with zero.

Here is my working
https://imgur.com/a/Maca4gH

Thanks
2. Would also like to know other ways to work out the derminant. The methods I know are first breaking them down into smaller 3*3 matrices like I did above, second using elementary row operation. Are there any other shorter method to deduce the answer quickly and accurately? Thanks.
3. (Original post by coconut64)
Would also like to know other ways to work out the derminant. The methods I know are first breaking them down into smaller 3*3 matrices like I did above, second using elementary row operation. Are there any other shorter method to deduce the answer quickly and accurately? Thanks.

Row/column operations and taking out factors would be my first choice. Finding the best operation can be a little time consuming.

In this case 6xrow2 added to row3, makes rows 1 and 3 almost identical. Then subtract row3 from row1, and you have a row with 3 zeroes in it, so expand along that row, and so on.
4. Thanks for the help. I used the generic method (I am not sure what it is called) for this question. However I obtained a wrong answer, but I am not sure where it had gone wrong...

Is the goal of elemetary row/ column operation in this example trying to achieve as many zeros as we can so that the expansion part is easier ?

ghostwalker
5. Can either use row or column operations as mentioned above to make it workable. If you're stuck for options do it the old fashioned way, but you will need to account for 3 different 3x3 matrix determinants, which makes life messy.
6. (Original post by coconut64)
Is the goal of elemetary row/ column operation in this example trying to achieve as many zeros as we can so that the expansion part is easier ?
ghostwalker
Yes, in particular maximise the zeroes in one row or one column.

Row/column operatrions are particularly useful with matrices where you can find/produce two rows of equal value, or two columns of equal value - in that case the determinant is zero, and no further working is required.

The initial line of your working producing the 3x3 determinants is correct. So, your error lines in evaluating them.
7. thotproduct
Hi thanks for helping! That's what I did in my calculation. I really don't understand in which part my calculation had gone wrong. Thanks
8. (Original post by coconut64)
However I obtained a wrong answer, but I am not sure where it had gone wrong...
Penultimate line should be -94 + 114 +8
9. ghostwalker

I can't seem to figure out how to reply in this new student room system but anyways.
https://imgur.com/a/rdkS9gl
Appreciate the help, I managed to spot my mistake in the first go after taking your hint. I have just tried applying the elementary operation to turn the matrix into echelon form. However now I am getting -28 instead of 28.

It seems as though using e.r.o is more prone to inaccuracy...
10. (Original post by coconut64)
ghostwalker

I can't seem to figure out how to reply in this new student room system but anyways.
https://imgur.com/a/rdkS9gl
Appreciate the help, I managed to spot my mistake in the first go after taking your hint. I have just tried applying the elementary operation to turn the matrix into echelon form. However now I am getting -28 instead of 28.

It seems as though using e.r.o is more prone to inaccuracy...
If you exchange two rows or two columns, that multiplies the determinant by -1.
11. But if I didn't know the answer, say this is the first method I use without checking with other methods. How do I know that -28 is the wrong answer considering I did change it into the echelon form...

I have changed it to the right echeon form?Thanks

ghostwalker
12. (Original post by coconut64)
But if I didn't know the answer, say this is the first method I use without checking with other methods. How do I know that -28 is the wrong answer considering I did change it into the echelon form...

I have changed it to the right echeon form?Thanks

ghostwalker
Echelon form is rather overkill, and not required.

You know -28 is wrong because you swapped two rows over and failed to multiply the determinant by -1 to counter the effects of that.
You need to check what effect e.r.o.s have on a determinant.

And don't get hung up on just row operations; column operations are equally valid when working out a determinant.
13. ghostwalker

Oh I see! My lecture notes omitted this rule; I genuinely had no clue that the multiplication of -1 is required when swapping rows. Thanks for the help!
Also would like to say a massive thank you for helping students out on Student Room. I remember receiving help from you I think in year 12 and 13 for all sorts of questions ranging from M1 to C4; although it has been quite a while ago (now I am in second year at uni doing Econ), the immediate support from the members had honestly been my major source of academic support at the time. So very glad to see that there are still people out there who are making such a significant contribution for others in need.

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Updated: January 12, 2019
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