Problem on computing the determinant of a 3x3 matrix
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Compute the determinant of $$\begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix}$$
Solution
Recall that
The method of diagonals for computing the determinant of a 3x3 matrix
The determinant of a $3 \times 3$ matrix can be computing by adding the products of terms on the forward diagonals and subtracting the products of terms on the backward diagonals.
That is, $$ \text{det} \begin{pmatrix} a&b&c\\d&e&f\\g&h&i \end{pmatrix} = aei + bfg + cdh - afh - bdi - ceg.$$
The determinant of $ \begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix}$ is thus \begin{align} 1\cdot 1 \cdot 4 &+ (-1) \cdot 1 \cdot 1 + 1\cdot 1\cdot 2 \\ &- 1\cdot 1 \cdot 1 - (-1) \cdot 1 \cdot 4 - 1 \cdot 1 \cdot 2,\end{align}
which simplifies to $$\text{det } \begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix} = 6.$$
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