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So I have here a matrix $X =$ \begin{bmatrix}3&0&-1&0\\0&2&-4&0\\0&-4&7&0\\-3&0&1&1\end{bmatrix}

The questions are:

  • (a) Use cofactor expansion to compute the determinant of $X$
  • (b) Using properties of determinants, compute $\det X.$

I was already able to compute for $\det X$ using cofactor expansion but I'm having trouble with $b.$ I already looked at the properties of determinants and I can't seem to find any properties that I could apply to solve for it. Could you guys help me how to approach this problem?

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If you replace the fourth row with its sum with the first row, you will get another matrix with the same determinant, but now that determinant is much easier to compute.

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  • $\begingroup$ I can't seem to determine what property is this? $\endgroup$ – user831749 Oct 3 '20 at 7:43
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    $\begingroup$ It's the proporty that it says that if you add to a row (or a column) a multiple of another row (or column), that will not change the determinant. It's property 14 from this list. $\endgroup$ – José Carlos Santos Oct 3 '20 at 7:50
  • $\begingroup$ I see. Thank you so much! $\endgroup$ – user831749 Oct 3 '20 at 7:52
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Use the following properties:

  • Adding a row (or column) of X multiplied by a scalar $k$ to another row (or column) of $X$, then the determinant will not change.
  • Triangle property: If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.

See 1 and 2 for more properties.

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  • $\begingroup$ So the first property you mentioned is just saying that if you do elementary row operations on a matrix, it doesn't change its determinant? Is it right that you could do multiple and the property still holds? Also, I don't understand how will I able to apply the triangle property? Should I first perform row operations to get to the point where "all the elements of a determinant above or below the main diagonal consist of zeros"? $\endgroup$ – user831749 Oct 3 '20 at 8:12
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    $\begingroup$ Yes, you apply the triangle property after the row operations. See this for the proof of the first property. $\endgroup$ – Äres Oct 3 '20 at 8:24

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