# Math Help - Determinant calculation. 4 Questions.

1. ## Determinant calculation. 4 Questions.

Need to calculate these matrix determinants without using any calculation programs, only by definition!

Thanks a lot to whom accept this challenge!

1. Above C

0 , 3i , 1
2-i , 3 , 1
-i , 0 , 2i-1

2. Matrix in which the diagonal is "a" and all the rest is "b"

a , b, ... , b
b , a,
.
.
.
b , ... , a

3. Matrix in which the diagonal is "0". Above the diagonal is "-1" and "1" is below the diagonal. (Need to separate to 2 cases when n is odd and when n is even)

0,-1,...,-1
1,0
.
.
.
1,... 0

4.

1,2,...,n
2,3,...,n+1
.
.
.
n,n+1,...,2n-1

2. Originally Posted by Also sprach Zarathustra
Need to calculate these matrix determinants without using any calculation programs, only by definition!

Thanks a lot to whom accept this challenge!

1. Above C

0 , 3i , 1
2-i , 3 , 1
-i , 0 , 2i-1
Just expand on, say, the first row:
$\left|\begin{array}{ccc}0 & 3i & 1 \\ 2-i & 3 & 1 \\ -i & 0 & 2i-1\end{array}\right|= -3i\begin\left|{array}{cc}2-i & 1 \\ -i & 2i-1\end{array}\right|+ \left|\begin{array}{cc}2- i & 3 \\ -i & 0\end{array}\right|$

For the rest, I recommend looking at the 2 by 2 and 3 by 3 versions and trying to see a pattern.
2. Matrix in which the diagonal is "a" and all the rest is "b"

a , b, ... , b
b , a,
.
.
.
b , ... , a

3. Matrix in which the diagonal is "0". Above the diagonal is "-1" and "1" is below the diagonal. (Need to separate to 2 cases when n is odd and when n is even)

0,-1,...,-1
1,0
.
.
.
1,... 0

4.

1,2,...,n
2,3,...,n+1
.
.
.
n,n+1,...,2n-1