Compute the determinant of the following matrix by expanding as specified. Where i and j are the position indices of the elements of the matrix. The answers in a, b, and c should be the same.

[3 4 -9; 2 10 3; 5 -2 6]

a) Along the first row:
b) Along the second row:
c) Along the third row:

Question

Compute the determinant of the following  matrix by expanding as specified. Where i and j are the position indices of the elements of the matrix. The answers  in a, b, and c should be the same.

[3 4 -9; 2 10 3; 5 -2 6]

a) Along the first row:
b) Along the second row:
c) Along the third row:

anonymous · Answer

Your matrix:
$$\begin{bmatrix}3&4&-9\2&10&3\5&-2&6\end{bmatrix}$$
I will the determinant along the second row, and the other two are identical, and up to you. First, we compute the cofactors along the second row:

$$A_{2,1}=(-1)^{2+1}\begin{vmatrix}4&-9\-2&6\end{vmatrix}=(-1)\left((4)(6)-(-9)(-2)\right)=-6$$

$$A_{2,2}=(-1)^{2+2}\begin{vmatrix}3&-9\5&6\end{vmatrix}=(+1)\left((3)(6)-(-9)(5)\right)=63$$

$$A_{2,3}=(-1)^{2+3}\begin{vmatrix}3&4\5&-2\end{vmatrix}=(-1)\left((3)(-2)-(5)(4)\right)=26$$

$$\det(A)=2(-6)+10(63)+3(26)=696$$
The other expansions will yield the same result.