List of top Mathematics Questions on Properties of Determinants

If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

If for some $c < 0$, the quadratic equation, $2cx^{2}-2\left(2c-1\right)x+3c^{2}=0$ has two distinct real roots, $\frac{1}{a}$ and $\frac{1}{b},$ then the value of the determinant. $\begin{vmatrix}1+a&1&1\\ 1&1+b&1\\ 1&1&1+c\end{vmatrix}$ is :

If the matrix beginbmatrix 1 & 3 & λ+2
2 & 4 & 8
3 & 5 & 10 endbmatrix is singular, then λ=

If [x] denotes the greatest integer ≤ x and -1≤ x<0, 0≤ y<1, 1≤ z<2, then the value of the determinant beginvmatrix [x]+1 & [y] & [z]
[x] & [y]+1 & [z]
[x] & [y] & [z]+1 endvmatrix is

If \(x, y, z\) are complex numbers, and
\[ \Delta = \begin{vmatrix} 0 & -y & -z \\ \overline{y} & 0 & -x \\ \overline{z} & \overline{x} & 0 \end{vmatrix}, \]
then \(\Delta\) is:

Let \(M\) be a \(3\times3\) non-singular matrix with \(\det(M)=\alpha\). If \(|M^{-1}\operatorname{adj}(M)|=K\), then the value of \(K\) is

If \[ \begin{vmatrix} p & q - y & r - z \\ p - x & q & r - z \\ p - x & q - y & r \end{vmatrix} = 0, \] then the value of \(\dfrac{p}{x} + \dfrac{q}{y} + \dfrac{r}{z}\) is

The determinant \[ \left| \begin{matrix} 1 & x & x^2 \\ 1 & x^3 & x^4 \\ 1 & x^5 & x^6 \end{matrix} \right| \] vanishes for

The value of the determinant \[ \begin{vmatrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 191 & 198 & 181 \end{vmatrix} \] is:

The value of vmatrix 1 & 2 & 3
-4 & 3 & 6
2 & -7 & 9 vmatrix is