# Thread: det(A) where A is 3x3 matrix

1. ## det(A) where A is 3x3 matrix [SOLVED]

If A is 3x3 with all entries 1's and 0's, what is the max value of det(A)?

I answered it to be 1, which is wrong. Can someone help me?

Thanks!

2. ## Re: det(A) where A is 3x3 matrix

Originally Posted by page929
If A is 3x3 with all entries 1's and 0's, what is the max value of det(A)?

I answered it to be 1, which is wrong. Can someone help me?

Thanks!
Hint:

3. ## Re: det(A) where A is 3x3 matrix

$\displaystyle A=\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\end{bmatrix}$

Use Laplace's formula on the first row and you'll get

$\displaystyle \det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)$

since each entry can only be either 1 or 0, just try to find the coefficients such that you are always summing and not subtracting. You will find that the answer to your question is 2.

4. ## Re: det(A) where A is 3x3 matrix

[1 1 0]
[0 1 1]
[1 0 1] maximizes the positive contributions, and minimizes the negative contributions. (seeing is believing).