# Thread: Determinants Question What did I do Wrong?

1. ## Determinants Question What did I do Wrong?

Does anyone know why this is wrong? I listed step by step what I did.

Find the following determinants by cofactor expansion along a convenient row or columns

|5 -1 5 0 -3 |
|4 0 3 -2 2 |
|3 2 1 0 3 |
|7 0 -3 0 4 |
|2 -1 4 3 -2 |

First, I took R5 ->2R5+3R2
I then got
|-5 -1 5 -0 -3|
|4 0 4 -2 2|
|3 2 1 0 3 |
|7 0 -3 0 4 |
|16-2 17 0 2 |

I then got -2 as a cofactor and used that as a pivot so i have -2 * the matrix below.
|-5 -1 5 -0 -3|
|3 2 1 0 3 |
|7 0 -3 0 4 |
|16-2 17 0 2 |

I then performed the following row operations:
R1-> 2R1 + R2
R4 -> R4 + R2 and I get
|-7 0 11 -3|
|3 2 1 3|
|7 0 -3 4|
|19 0 18 5|

I used the row 3,2,1,3 as the pivot and I have the resulting matrix
-4 cofactor times
|-7 11 -3|
|7 -3 4|
|19 18 5|

If I just take the value of the determinant 3 x 3 I get the correct answer, but shouldn't i multiply the matrix by -4? Intuitively I feel like it should be multiplied by -5. Any thoughts?

2. let $A=$ $\begin{bmatrix}5&-1&5&0&-3\\4&0&3&-2&2\\3&2&1&0&3\\7&0&-3&0&4\\2&-1&4&3&-2\end{bmatrix}$ $\frac{3}{2}R_{2}+R_{5}\rightarrow{R_{5}}$ $\begin{bmatrix}5&-1&5&0&-3\\4&0&3&-2&2\\3&2&1&0&3\\7&0&-3&0&4\\8&-1&\frac{17}{2}&0&1\end{bmatrix}$

As you can see, we can expand from the -2 (row 2 column 4)...

$(-1)^{6}(-2)$ $\begin{bmatrix}5&-1&5&-3\\3&2&1&3\\7&0&-3&4\\8&-1&\frac{17}{2}&1\end{bmatrix}$

Now we have a 4X4 matrix to reduce...

$\begin{bmatrix}5&-1&5&-3\\3&2&1&3\\7&0&-3&4\\8&-1&\frac{17}{2}&1\end{bmatrix}$ $(-1)R_{1}+R_{4}\rightarrow{R_{4}}$ $\begin{bmatrix}5&-1&5&-3\\3&2&1&3\\7&0&-3&4\\3&0&\frac{7}{2}&4\end{bmatrix}$ $2R_{1}+R_{2}\rightarrow{R_{2}}$ $\begin{bmatrix}5&-1&5&-3\\13&0&11&-3\\7&0&-3&4\\3&0&\frac{7}{2}&4\end{bmatrix}$

Now, expand from -1 (row 1 column 2)...

We have...

$(-1)^{3}(-1)\begin{bmatrix}13&11&-3\\7&-3&4\\3&\frac{7}{2}&4\end{bmatrix}$

Then...

$det(A)=$ $\begin{bmatrix}13&11&-3\\7&-3&4\\3&\frac{7}{2}&4\end{bmatrix}$ $\begin{matrix}13&11\\7&-3\\3&\frac{7}{2}\end{matrix}$

Thus, $det(A)=(-2)[(-156)+(132)+(\frac{-147}{2})-(27)-(182)-(308)]=1229$

(the -2 comes from our two previous cofactors)

Which agrees with my Ti-89!

3. Hmm I don't think this is correct according to the answer in the back and my calculator the answer should be -511 but I can't seem to figure out why