If the system of linear equations 3x + y + beta z = 3$$3x + y + \beta z = 3$$ 2x + alpha y - z = -3$$2x + \alpha y - z = -3$$ x + 2y + z = 4$x + 2y + z = 4$ has infinitely many solutions, then the value of 22beta - 9alpha$22\beta - 9\alpha$ is:

### Related Formula For a matrix M$M$ of order k$k$: det(cM) = c^k det(M)$$\det(cM) = c^k \det(M)$$ det(textadj(M)) = (det(M))^k-1$$\det(\text{adj}(M)) = (\det(M))^{k-1}$$ ### 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 A = beginbmatrix 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 endbmatrix - beginbmatrix 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 endbmatrix = beginbmatrix 0 & a & 1 \\ 2 & 0 & 0 \\ a & 1 & 1 endbmatrix$$A = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & a & 1 \\ 2 & 0 & 0 \\ a & 1 & 1 \end{bmatrix}$$ Evaluate det(A)$\det(A)$ by expanding down the second row: det(A) = -2(a - 1) = 2 - 2a$$\det(A) = -2(a - 1) = 2 - 2a$$ Given det(A) = -4$\det(A) = -4$: 2 - 2a = -4 implies 2a = 6 implies a = 3$$2 - 2a = -4 \implies 2a = 6 \implies a = 3$$ ### Step 2: Substitute constant and expand targeting expression For a = 3$a = 3$, the target expression becomes: det((3+1)textadj((3-1)A)) = det(4textadj(2A))$$\det((3+1)\text{adj}((3-1)A)) = \det(4\text{adj}(2A))$$ Since the matrix order is 3 times 3$3 \times 3$: det(4textadj(2A)) = 4^3 det(textadj(2A)) = 64 (det(2A))^3-1 = 64 (det(2A))^2$$\det(4\text{adj}(2A)) = 4^3 \det(\text{adj}(2A)) = 64 (\det(2A))^{3-1} = 64 (\det(2A))^2$$ Now unpack det(2A)$\det(2A)$: det(2A) = 2^3 det(A) = 8(-4) = -32$$\det(2A) = 2^3 \det(A) = 8(-4) = -32$$ Substitute this value back: textTotal Determinant = 64 times (-32)^2 = 2^6 times (2^5)^2 = 2^6 times 2^10 = 2^16$$\text{Total Determinant} = 64 \times (-32)^2 = 2^6 \times (2^5)^2 = 2^6 \times 2^{10} = 2^{16}$$ ### Step 3: Alternative calculation matching shifts Following the alternate parsing blueprint: det(4textadj(2A)) = 4^3 cdot 2^2(3-1) cdot 3^2(3-1) cdot |A|^2 = 2^6 cdot 3^6 cdot (-4)^2 = 2^10 cdot 3^6$$\det(4\text{adj}(2A)) = 4^3 \cdot 2^{2(3-1)} \cdot 3^{2(3-1)} \cdot |A|^2 = 2^6 \cdot 3^6 \cdot (-4)^2 = 2^{10} \cdot 3^6$$ Comparing exponents to 2^m cdot 3^n$2^m \cdot 3^n$: m = 10, quad n = 6 implies m + n = 16$$m = 10, \quad n = 6 \implies m + n = 16$$ ### 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$\det$, and once inside when computing the sub-adjoint scaling factor. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants

### Related Formula Using standard matrix properties: - M \, textadj(M) = |M|I$M \, \text{adj}(M) = |M|I$ - textadj(kM) = k^n-1textadj(M)$\text{adj}(kM) = k^{n-1}\text{adj}(M)$ - |M^k| = |M|^k$|M^k| = |M|^k$ - [textadj(M^-1)] = [textadj(M)]^-1$[\text{adj}(M^{-1})] = [\text{adj}(M)]^{-1}$ ### Core Logic First, find lambda$\lambda$ by evaluating |A| = -1$|A| = -1$: |A| = lambda(10 - (-6)) - 2(8 - 42) + 3(-4 - 35) = -1$$|A| = \lambda(10 - (-6)) - 2(8 - 42) + 3(-4 - 35) = -1$$ 16lambda - 2(-34) + 3(-39) = -1$$16\lambda - 2(-34) + 3(-39) = -1$$ 16lambda + 68 - 117 = -1 implies 16lambda - 49 = -1 implies 16lambda = 48 implies lambda = 3$$16\lambda + 68 - 117 = -1 \implies 16\lambda - 49 = -1 \implies 16\lambda = 48 \implies \lambda = 3$$ ### Step 1: Simplifying matrix expression B$B$ Let C = A \, textadj(A^2)$C = A \, \text{adj}(A^2)$. We know: A^2 \, textadj(A^2) = |A^2|I = |A|^2 I$$A^2 \, \text{adj}(A^2) = |A^2|I = |A|^2 I$$ Multiply C$C$ by A$A$ on the left: AC = A^2 \, textadj(A^2) = |A|^2 I$$AC = A^2 \, \text{adj}(A^2) = |A|^2 I$$ Since |A| = -1 implies |A|^2 = 1$|A| = -1 \implies |A|^2 = 1$: AC = I implies C = A^-1$$AC = I \implies C = A^{-1}$$ Now, we are given B^-1 = textadj(C) = textadj(A^-1)$B^{-1} = \text{adj}(C) = \text{adj}(A^{-1})$: B = [textadj(A^-1)]^-1 = textadj(A)$$B = [\text{adj}(A^{-1})]^{-1} = \text{adj}(A)$$ ### Step 2: Determinant calculation of lambda B + I$\lambda B + I$ We need to find |lambda B + I| = |3textadj(A) + I|$|\lambda B + I| = |3\text{adj}(A) + I|$: Let P = 3textadj(A) + I$P = 3\text{adj}(A) + I$. AP = 3Atextadj(A) + A = 3|A|I + A = -3I + A = A - 3I$$AP = 3A\text{adj}(A) + A = 3|A|I + A = -3I + A = A - 3I$$ Taking determinants on both sides: |A| cdot |P| = |A - 3I| implies -|P| = |A - 3I| implies |P| = -|A - 3I|$$|A| \cdot |P| = |A - 3I| \implies -|P| = |A - 3I| \implies |P| = -|A - 3I|$$ Let's calculate matrix A - 3I$A - 3I$: A - 3I = beginbmatrix 0 & 2 & 3 \\ 4 & 2 & 6 \\ 7 & -1 & -1 endbmatrix$$A - 3I = \begin{bmatrix} 0 & 2 & 3 \\ 4 & 2 & 6 \\ 7 & -1 & -1 \end{bmatrix}$$ textDet(A-3I) = 0 - 2(4(-1) - 42) + 3(-4 - 14) = -2(-46) + 3(-18) = 92 - 54 = 38$$\text{Det}(A-3I) = 0 - 2(4(-1) - 42) + 3(-4 - 14) = -2(-46) + 3(-18) = 92 - 54 = 38$$ Thus: |P| = -38$|P| = -38$ Taking absolute value of determinant / magnitude: ||lambda B + I|| = 38$$||\lambda B + I|| = 38$$ ### Pattern Recognition For products of adjoint matrices, always relate back to basic identity M \, textadj(M) = |M|I$M \, \text{adj}(M) = |M|I$. Pre-multiplying by A$A$ or A^2$A^2$ collapses long chains of adjoints instantly into standard scalar multiples. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants

### Related Formula For any n times n$n \times n$ matrix A$A$, the determinant properties of adjoints scale iteratively as follows: |textadj A| = |A|^n-1$$|\text{adj } A| = |A|^{n-1}$$ |textadj(adj A)| = |A|^(n-1)^2$$|\text{adj(adj } A)| = |A|^{(n-1)^2}$$ |textadj(adj(adj A))| = |A|^(n-1)^3$$|\text{adj(adj(adj } A))| = |A|^{(n-1)^3}$$ ### Core Logic Since A$A$ is a 3 times 3$3 \times 3$ matrix (n=3$n=3$): |textadj(adj(adj A))| = |A|^(3-1)^3 = |A|^8 = 81$$|\text{adj(adj(adj } A))| = |A|^{(3-1)^3} = |A|^8 = 81$$ |A|^8 = 3^4 implies |A|^2 = 3 implies |A| = 3^1/2 = sqrt3$$|A|^8 = 3^4 \implies |A|^2 = 3 \implies |A| = 3^{1/2} = \sqrt{3}$$ Now look at the power base for the equation: |textadj(adj A)| = |A|^(3-1)^2 = |A|^4$|\text{adj(adj } A)| = |A|^{(3-1)^2} = |A|^4$. Substitute this into the matching requirement equation set: left(|A|^4right)^frac(n-1)^22 = |A|^3n^2 - 5n - 4$$\left(|A|^4\right)^{\frac{(n-1)^2}{2}} = |A|^{3n^2 - 5n - 4}$$ |A|^2(n-1)^2 = |A|^3n^2 - 5n - 4$$|A|^{2(n-1)^2} = |A|^{3n^2 - 5n - 4}$$ Equating exponents since bases are identical: 2(n - 1)^2 = 3n^2 - 5n - 4$$2(n - 1)^2 = 3n^2 - 5n - 4$$ 2(n^2 - 2n + 1) = 3n^2 - 5n - 4$$2(n^2 - 2n + 1) = 3n^2 - 5n - 4$$ 2n^2 - 4n + 2 = 3n^2 - 5n - 4$$2n^2 - 4n + 2 = 3n^2 - 5n - 4$$ n^2 - n - 6 = 0$n^2 - n - 6 = 0$ ### Step 1: Solve for Exponent Parameter Factoring the quadratic parameter relation: (n - 3)(n + 2) = 0 implies n = 3 quad textor quad n = -2$$(n - 3)(n + 2) = 0 \implies n = 3 \quad \text{or} \quad n = -2$$ Both choices are valid integers, so the set S = \-2, 3\$S = \{-2, 3\}$. ### Step 2: Calculate the Target Summation We need to evaluate sum_nin S |A^n^2 + n| = |A^(-2)^2 + (-2)| + |A^(3)^2 + 3|$\sum_{n\in S} |A^{n^2 + n}| = |A^{(-2)^2 + (-2)}| + |A^{(3)^2 + 3}|$: - For n = -2$n = -2$, n^2 + n = 4 - 2 = 2 implies |A^2| = |A|^2 = 3$n^2 + n = 4 - 2 = 2 \implies |A^2| = |A|^2 = 3$ - For n = 3$n = 3$, n^2 + n = 9 + 3 = 12 implies |A^12| = |A|^12 = (sqrt3)^12 = 3^6 = 729$n^2 + n = 9 + 3 = 12 \implies |A^{12}| = |A|^{12} = (\sqrt{3})^{12} = 3^6 = 729$ Summing these evaluated values: textTotal = 3 + 729 = 732$$\text{Total} = 3 + 729 = 732$$ ### Pattern Recognition Always remember that |A^k| = |A|^k$|A^k| = |A|^k$. Calculating determinant transformations directly as scalar power factors first prevents rendering high order numerical values prematurely. ### Evaluation Rubric / Model Answer null ### 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 Delta$\Delta$ must equal 0$0$, along with all auxiliary determinants: Delta_x = Delta_y = Delta_z = 0$\Delta_x = \Delta_y = \Delta_z = 0$. The perpendicular radius distance from a point (x_0, y_0)$(x_0, y_0)$ to a line Ax + By + C = 0$Ax + By + C = 0$ is: d = frac|Ax_0 + By_0 + C|sqrtA^2 + B^2$$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$ ### Core Logic Set the coefficient matrix determinant Delta = 0$\Delta = 0$: left| beginmatrix 2 & 3 & 5 \\ 7 & 3 & -2 \\ 12 & 3 & -(lambda + 4) endmatrix right| = 0$$\left| \begin{matrix} 2 & 3 & 5 \\ 7 & 3 & -2 \\ 12 & 3 & -(\lambda + 4) \end{matrix} \right| = 0$$ Expanding row layout variations or applying column actions yields: 2[-3(lambda + 4) + 6] - 3[-7(lambda + 4) + 24] + 5[21 - 36] = 0$$2[-3(\lambda + 4) + 6] - 3[-7(\lambda + 4) + 24] + 5[21 - 36] = 0$$ -6lambda - 24 + 12 + 21lambda + 84 - 72 - 75 = 0$$-6\lambda - 24 + 12 + 21\lambda + 84 - 72 - 75 = 0$$ 15lambda - 75 = 0 implies lambda = 5$$15\lambda - 75 = 0 \implies \lambda = 5$$ ### Step 1: Solve for the Constant Parameter Using Cramer's condition for the infinite solution constraint, evaluate the structural augmented row columns component matrix Delta_y = 0$\Delta_y = 0$: left| beginmatrix 2 & 9 & 5 \\ 7 & 8 & -2 \\ 12 & 16 - mu & -9 endmatrix right| = 0$$\left| \begin{matrix} 2 & 9 & 5 \\ 7 & 8 & -2 \\ 12 & 16 - \mu & -9 \end{matrix} \right| = 0$$ Solving this configuration yields: mu = 9$\mu = 9$ Hence, the circle center is (lambda, mu) = (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: textRadius = frac|4(5) - 3(9)|sqrt4^2 + (-3)^2 = frac|20 - 27|5 = frac|-7|5 = frac75$$\text{Radius} = \frac{|4(5) - 3(9)|}{\sqrt{4^2 + (-3)^2}} = \frac{|20 - 27|}{5} = \frac{|-7|}{5} = \frac{7}{5}$$ ### Pattern Recognition Notice that since the y-coefficients are identical (3, 3, 3$3, 3, 3$) across all lines, performing raw row subtractions (R_2 - R_1$R_2 - R_1$ and R_3 - R_2$R_3 - R_2$) strips away the y$y$-variable instantly during structural matrix processing. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Straight Lines

### Related Formula A 2 times 2$2 \times 2$ matrix left[ beginmatrix a & d \\ b & c endmatrix right]$\left[ \begin{matrix} a & d \\ b & c \end{matrix} \right]$ is singular if its determinant equals 0$0$: ad - bc = 0 implies ad = bc$$ad - bc = 0 \implies ad = bc$$ ### 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 times 2 = 2 times 2$2 \times 2 = 2 \times 2$. Since there are 4 distinct choices in S$S$, this provides: textWays = ^4C_1 = 4$$\text{Ways} = ^4C_1 = 4$$ ### 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: ^4C_2 = 6$^4C_2 = 6$ pairs. Each pair can be arranged in 4 unique layouts (2 times 2$2 \times 2$ arrangements). textWays = 6 times 4 = 24$$\text{Ways} = 6 \times 4 = 24$$ ### 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. textWays = 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 times 9 = 18$2 \times 9 = 18$ and 3 times 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: textWays = 2 times 4 = 8$$\text{Ways} = 2 \times 4 = 8$$ ### Step 3: Total Summation Sum the total count of valid singular configurations: textTotal Matrices = 4 + 24 + 0 + 8 = 36$$\text{Total Matrices} = 4 + 24 + 0 + 8 = 36$$ ### 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. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Permutations and Combinations