Let A$A$ be a 3 × 3$3 \times 3$ matrix such that |adj(adj(adj A))| = 81.$$|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))| = 81.$$ If S = n in Z : (|adj(adj A)|)((n - 1)²)/(2) = |A|3n² - 5n - 4$S = \left\{n \in \mathbb{Z} : \left(|\operatorname{adj}(\operatorname{adj} A)|\right)^{\frac{(n - 1)^2}{2}} = |A|^{3n^2 - 5n - 4}\right\}$, then Σn in S |An² + n|$\sum_{n \in S} |A^{n^2 + n}|$ is equal to

Related Formula

For a matrix M$M$ of order k$k$:

Core Logic

Isolate matrix A$A$ from the given expression, calculate its parameter value a$a$ using the determinant value constraint, and simplify the adjoint property expression step-by-step.

Step 1: Isolate and evaluate determinant of A

Evaluate (A)$\det(A)$ by expanding down the second row:

Given (A) = -4$\det(A) = -4$:

Step 2: Substitute constant and expand targeting expression

For a = 3$a = 3$, the target expression becomes:

Since the matrix order is 3 × 3$3 \times 3$:

Now unpack (2A)$\det(2A)$:

Substitute this value back:

Step 3: Alternative calculation matching shifts

Following the alternate parsing blueprint:

Comparing exponents to 2^m · 3ⁿ$2^m \cdot 3^n$:

Pattern Recognition

Be careful when factoring scalar coefficients out of an adjoint expression—the dimension exponent applies twice: once for the scalar prefix out of $\det$, and once inside when computing the sub-adjoint scaling factor.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

Related Formula

Using standard matrix properties:

• M adj(M) = |M|I$M \, \text{adj}(M) = |M|I$
• adj(kM) = kⁿ⁻¹adj(M)$\text{adj}(kM) = k^{n-1}\text{adj}(M)$
• |M^k| = |M|^k$|M^k| = |M|^k$
• [adj(M⁻¹)] = [adj(M)]⁻¹$[\text{adj}(M^{-1})] = [\text{adj}(M)]^{-1}$

Core Logic

First, find λ$\lambda$ by evaluating |A| = -1$|A| = -1$:

Step 1: Simplifying matrix expression B$B$

Let C = A adj(A²)$C = A \, \text{adj}(A^2)$. We know:

Multiply C$C$ by A$A$ on the left:

Since |A| = -1 |A|² = 1$|A| = -1 \implies |A|^2 = 1$:

Now, we are given B⁻¹ = adj(C) = adj(A⁻¹)$B^{-1} = \text{adj}(C) = \text{adj}(A^{-1})$:

Step 2: Determinant calculation of λ B + I$\lambda B + I$

We need to find |λ B + I| = |3adj(A) + I|$|\lambda B + I| = |3\text{adj}(A) + I|$: Let P = 3adj(A) + I$P = 3\text{adj}(A) + I$.

Taking determinants on both sides:

Let's calculate matrix A - 3I$A - 3I$:

Thus: |P| = -38$|P| = -38$ Taking absolute value of determinant / magnitude:

Pattern Recognition

For products of adjoint matrices, always relate back to basic identity M adj(M) = |M|I$M \, \text{adj}(M) = |M|I$. Pre-multiplying by A$A$ or A²$A^2$ collapses long chains of adjoints instantly into standard scalar multiples.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants

Related Formula

For a system of three linear equations to possess infinitely many solutions, the main determinant Δ$\Delta$ must equal 0$0$, along with all auxiliary determinants: Δₓ = Δy = Δz = 0$\Delta_x = \Delta_y = \Delta_z = 0$. The perpendicular radius distance from a point (x₀, y₀)$(x_0, y_0)$ to a line Ax + By + C = 0$Ax + By + C = 0$ is:

Core Logic

Set the coefficient matrix determinant Δ = 0$\Delta = 0$:

Expanding row layout variations or applying column actions yields:

Step 1: Solve for the Constant Parameter

Using Cramer's condition for the infinite solution constraint, evaluate the structural augmented row columns component matrix Δy = 0$\Delta_y = 0$:

Solving this configuration yields: μ = 9$\mu = 9$

Hence, the circle center is (λ, μ) = (5, 9)$(\lambda, \mu) = (5, 9)$.

Step 2: Calculate Radius via Distance

The line is given as 4x - 3y = 0$4x - 3y = 0$. Calculate the perpendicular distance from (5, 9)$(5, 9)$ to this line:

Pattern Recognition

Notice that since the y-coefficients are identical (3, 3, 3$3, 3, 3$) across all lines, performing raw row subtractions (R₂ - R₁$R_2 - R_1$ and R₃ - R₂$R_3 - R_2$) strips away the y$y$-variable instantly during structural matrix processing.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Straight Lines

Related Formula

A 2 × 2$2 \times 2$ matrix [ matrix a & d b & c matrix ]$\left[ \begin{matrix} a & d \\ b & c \end{matrix} \right]$ is singular if its determinant equals 0$0$:

Core Logic

Elements must be chosen from the set S = 2, 3, 6, 9$S = \{2, 3, 6, 9\}$. We need to count all valid quadruplets (a, b, c, d)$(a, b, c, d)$ such that the product of the main diagonal equals the product of the off-diagonal.

Let's analyze systematic product matching cases:

• Case 1: Exactly 1 distinct number is used across all four slots.

Example: 2 × 2 = 2 × 2$2 \times 2 = 2 \times 2$. Since there are 4 distinct choices in S$S$, this provides:

Step 1: Evaluate Multi Number Configurations

• Case 2: Exactly 2 distinct numbers are used.

The numbers must pair up to provide identical products (e.g., a=d$a=d$ and b=c$b=c$). Choosing 2 numbers from 4: ⁴C₂ = 6$^4C_2 = 6$ pairs. Each pair can be arranged in 4 unique layouts (2 × 2$2 \times 2$ arrangements).

Step 2: Evaluate Four Distinct Element Sets

• Case 3: Exactly 3 distinct numbers are used.

None will satisfy the strict ad = bc$ad = bc$ zero determinant equality constraints without repeating products. Ways = 0$\text{Ways} = 0$

• Case 4: Exactly 4 distinct numbers are used.

We need ad = bc$ad = bc$ using the entire set 2, 3, 6, 9$\{2, 3, 6, 9\}$. Notice that 2 × 9 = 18$2 \times 9 = 18$ and 3 × 6 = 18$3 \times 6 = 18$. This forms a matching product combination pair. The number of unique matrices that can be built by assigning these elements to the diagonal slots is:

Step 3: Total Summation

Sum the total count of valid singular configurations:

Pattern Recognition

Always break down element counting problems involving determinants into distinct number-repetition subsets (1 number repeated, 2 numbers repeated, etc.) to ensure no configuration is missed.

Chapter Mix

Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Permutations and Combinations