Determinant Calculation
First, we evaluate the determinant of the given matrix:
Let $D = \begin{vmatrix} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{vmatrix}$. Expanding the determinant, we get: $D = x(yz - 1 \times 1) - 1(1 \times z - 1 \times 1) + 1(1 \times 1 - y \times 1)$ $D = x(yz - 1) - (z - 1) + (1 - y)$ $D = xyz - x - z + 1 + 1 - y$ $D = xyz - x - y - z + 2$
Applying Given Conditions
We are given that the determinant $D$ is positive:
$D > 0$
Substituting the calculated expression for $D$:
$xyz - x - y - z + 2 > 0$
Relating to p and q
We are also given the definitions $p = x + y + z$ and $q = xyz$. Substitute these into the inequality:
$q - (x + y + z) + 2 > 0$
$q - p + 2 > 0$
Final Inequality
Rearranging the terms to match the options:
$q + 2 > p$
This inequality matches Option C.
Important Questions from Matrices and Determinants
What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?
In obtaining the solution of the system of equations $x + y + z = 7$, $x + 2y+3z = 16$ and $x + 3y+4z = 22$ by Cramer's rule, the value of $y$ is obtained by dividing $D$ by $D_2$, where
$D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix}$
What is the value of the determinant $D_2$?
Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB)^{-1} = A^{-1}B^{-1}$
II. $(BA)(AB)^{-1} = I$, where $I$ is the identity matrix
III. $(AB)^T = A^T B^T$
How many of the above are correct?
Consider the following statements :
Statement-I :
If $X$ is an $n \times n$ matrix, then $\det(mX) = m^n \det(X)$, where $m$ is a scalar.
Statement-II :
If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det(Y) = m\det(X)$.
Which one of the following is correct in respect of the above statements?
Consider the following statements about the matrix $M = \begin{vmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{vmatrix}$
Statement-I : The inverse of $M$ does not exist.
Statement-II : $M$ is non-singular.
Which one of the following is correct in respect of the above statements?
