Answer:
Determinant are special number that can only be defined for square matrices.
Step-by-step explanation:
Determinant are particularly important for analysis. The inverse of a matrix exist, if the determinant is not equal to zero.
How to find determinant
For a 2×2 matrix
[tex]det ( \left[\begin{array}{cc}x&y\\a&z\end{array}\right] ) = xz-ay[/tex]
For a 3×3 matrix
we first decompose it to 2×2
[tex]det (\left[\begin{array}{ccc}k&l&m\\o&p&q\\r&s&t\end{array}\right] )\\\\= k*det(\left[\begin{array}{cc}p&q\\s&t\end{array}\right] ) - l*det(\left[\begin{array}{cc}o&q\\r&t\end{array}\right] ) + m*det(\left[\begin{array}{cc}o&p\\r&s\end{array}\right] ) \\\\=k(pt-sq) - l(ot-rq) + m(os-rp)[/tex]
Example
Find the values of λ for which the determinant is zero
[tex]\left[\begin{array}{ccc}s&-1&0\\-1&s&-1\\0&-1&1\end{array}\right][/tex]
[tex]det(\left[\begin{array}{ccc}s&-1&0\\-1&s&-1\\0&-1&1\end{array}\right])\\\\= s*det(\left[\begin{array}{cc}s&-1\\-1&1\end{array}\right] ) - (-1)*det(\left[\begin{array}{cc}-1&-1\\0&1\end{array}\right] ) + 0*det(\left[\begin{array}{cc}-1&s\\0&-1\end{array}\right] )\\\\= s(s(1)-(-1*-1)) - (-1)(-1*1 - (-1*0)) + 0\\= s(s - 1)) + 1(-1 + 0) \\=s^{2} -s-1[/tex]
Equating the determinant to zero
[tex]s^{2} -s-1 =0\\[/tex]
s = [tex]\frac{1}{2}[/tex] * (1 ±5 )
s = 1.61 or -0.61
