Finding the determinant of a $3 \times 3$ matrix via Laplace Expansion

182 Views Asked by At

I have a matrix here where I need to calculate the determinant using Laplace expansion.

$$ \begin{pmatrix} 4 & 0 & 1\\19 & 1 & -3\\7 & 1 & 0 \end{pmatrix} $$

So I did the simple diagonal & antidiagonal way to calculate and get

$$ \begin{align*} |A| &= 0 + 0 + 19-7-0-(-12) \\&=24 \end{align*}$$

Right now, I have to show that, by Laplace expansion, and am having a mistake:

$$ \begin{align*} |A| &= 4\begin{vmatrix}1 & -3\\1 & 0\end{vmatrix}+ 0\begin{vmatrix}19 & -3\\7 & 0\end{vmatrix}+ 1\begin{vmatrix}19 & 1\\7 & 1\end{vmatrix}\\ &= 4(-3) + 0 + 1(12)\\ &= 0 \end{align*}$$

Could someone help and spot my errors here?

UPDATE: The 2nd last row should be $3$, not $-3$, as shown in the answer.

1

There are 1 best solutions below

0
On BEST ANSWER

$$\left|\begin{array}{cc}1 & -3\\1 & 0\end{array}\right|=(1\cdot 0)-(1\cdot-3)=0-(-3)=3$$