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det(\left(\begin{matrix}i&j&k\\1&1&1\\1&1&-1\end{matrix}\right))
Find the determinant of the matrix using the method of diagonals.
\left(\begin{matrix}i&j&k&i&j\\1&1&1&1&1\\1&1&-1&1&1\end{matrix}\right)
Extend the original matrix by repeating the first two columns as the fourth and fifth columns.
-i+j+k=j+k-i
Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.
k+i-j=i+k-j
Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.
j+k-i-\left(i+k-j\right)
Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.
2j-2i
Subtract k+i-j from -i+j+k.
det(\left(\begin{matrix}i&j&k\\1&1&1\\1&1&-1\end{matrix}\right))
Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).
idet(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))-jdet(\left(\begin{matrix}1&1\\1&-1\end{matrix}\right))+kdet(\left(\begin{matrix}1&1\\1&1\end{matrix}\right))
To expand by minors, multiply each element of the first row by its minor, which is the determinant of the 2\times 2 matrix created by deleting the row and column containing that element, then multiply by the element's position sign.
i\left(-1-1\right)-j\left(-1-1\right)+k\left(1-1\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.
-2i-j\left(-2\right)
Simplify.
2j-2i
Add the terms to obtain the final result.