So i was given this question
Use determinants to find which real values of c make each of the following matrices invertible
$
\left[ {\begin{array}{cc}
0 & c & -c \\
-1 & 2& 1 \\
c & -c & -c
\end{array} } \right]
$
When i look at the solutions for this question they usually adding or subtracting a column or row to one another, until there is two consecutive $0s$ in the row or column then make a 2×2 matrix. I understand the logic behind finding a determinant, but for this question and similar ones is there a rule or method to go about solving these kinds of problems, and how to do it?
Best Answer
Note that if you multiply a row or column by a constant, then since the determinant is multi linear, the determinant of the resulting matrix is a constant times the original matrix.
Hence $\det \begin{bmatrix} 0 & c & -c \\ -1 & 2& 1 \\ c & -c & -c \end{bmatrix} = c \det \begin{bmatrix} 0 & 1 & -1 \\ -1 & 2& 1 \\ c & -c & -c \end{bmatrix} = c^2 \det \begin{bmatrix} 0 & 1 & -1 \\ -1 & 2& 1 \\ 1 & -1 & -1 \end{bmatrix}$.
We also have $\det \begin{bmatrix} 0 & 1 & -1 \\ -1 & 2& 1 \\ 1 & -1 & -1 \end{bmatrix} = 1$.