\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]
\( x + y \)
We are given the properties of the adjugate matrices of \( A \) and \( B \), and we need to determine \( \left( |Adj(AB)| \right)^{-1} \).
Step 1: Property of Determinant of Adjugate
For any square matrix \( M \), the determinant of the adjugate is given by: \[ |Adj(M)| = |M|^{n-1} \] where \( n \) is the order of the matrix.
Step 2: Using the Determinant Multiplication Rule
Since \( AB \) is the product of two matrices, we use the property: \[ |Adj(AB)| = |AB|^{n-1} \] Applying the determinant property: \[ |AB| = |A| \cdot |B| \] Thus, \[ |Adj(AB)| = (|A| \cdot |B|)^{n-1} \]
Step 3: Expressing in Terms of Given Values
We know that: \[ |Adj(A)| = |A|^{n-1} = x, \quad |Adj(B)| = |B|^{n-1} = y. \] Multiplying these equations: \[ |Adj(A)| \cdot |Adj(B)| = (|A|^{n-1}) \cdot (|B|^{n-1}) = |AB|^{n-1}. \] So, \[ |Adj(AB)| = |Adj(A)| \cdot |Adj(B)| = x \cdot y. \]
Step 4: Finding the Inverse
\[ \left( |Adj(AB)| \right)^{-1} = \frac{1}{|Adj(AB)|} = \frac{1}{xy}. \]
Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \).
\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]
The roots of the equation \( x^3 - 3x^2 + 3x + 7 = 0 \) are \( \alpha, \beta, \gamma \) and \( w, w^2 \) are complex cube roots of unity. If the terms containing \( x^2 \) and \( x \) are missing in the transformed equation when each one of these roots is decreased by \( h \), then
If \( a \neq b \neq c \), then
\[ \Delta_1 = \begin{vmatrix} 1 & a^2 & bc \\ 1 & b^2 & ca \\ 1 & c^2 & ab \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]and
\[ \frac{\Delta_1}{\Delta_2} = \frac{6}{11} \]then what is \( 11(a + b + c) \)?
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]