### Related Formula The quadratic form condition mathbfX^mathrmTmathbfAmathbfX = mathbf0$\mathbf{X}^{\mathrm{T}}\mathbf{A}\mathbf{X} = \mathbf{0}$ holds for all non-zero vectors mathbfX$\mathbf{X}$ if and only if A$A$ is a skew-symmetric matrix. For a 3 times 3$3 \times 3$ skew-symmetric matrix, the components satisfy: A = beginbmatrix 0 & x_1 & x_2 \\ -x_1 & 0 & x_3 \\ -x_2 & -x_3 & 0 endbmatrix$A = \begin{\bmatrix} 0 & x_1 & x_2 \\ -x_1 & 0 & x_3 \\ -x_2 & -x_3 & 0 \end{\bmatrix}$ ### Core Logic Let's define the matrix A$A$ using the parameters of a standard skew-symmetric form: A = beginbmatrix 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 endbmatrix$A = \begin{\bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{\bmatrix}$ Apply the first given matrix multiplication vector condition: A beginbmatrix 1 \\ 1 \\ 1 endbmatrix = beginbmatrix a+b \\ -a+c \\ -b-c endbmatrix = beginbmatrix 1 \\ 4 \\ -5 endbmatrix$A \begin{\bmatrix} 1 \\ 1 \\ 1 \end{\bmatrix} = \begin{\bmatrix} a+b \\ -a+c \\ -b-c \end{\bmatrix} = \begin{\bmatrix} 1 \\ 4 \\ -5 \end{\bmatrix}$ This gives the system of linear equations: a + b = 1 quad dots (1)$a + b = 1 \quad \dots (1)$ -a + c = 4 quad dots (2)$-a + c = 4 \quad \dots (2)$ -b - c = -5 implies b + c = 5 quad dots (3)$-b - c = -5 \implies b + c = 5 \quad \dots (3)$ Apply the second given matrix multiplication vector condition: A beginbmatrix 1 \\ 2 \\ 1 endbmatrix = beginbmatrix 2a+b \\ -a+c \\ -b-2c endbmatrix = beginbmatrix 0 \\ 4 \\ -8 endbmatrix$A \begin{\bmatrix} 1 \\ 2 \\ 1 \end{\bmatrix} = \begin{\bmatrix} 2a+b \\ -a+c \\ -b-2c \end{\bmatrix} = \begin{\bmatrix} 0 \\ 4 \\ -8 \end{\bmatrix}$ This gives the equation: 2a + b = 0 quad dots (4)$2a + b = 0 \quad \dots (4)$ ### Step 1: Solve for Matrix Elements Subtract equation (1) from equation (4): (2a + b) - (a + b) = 0 - 1 implies a = -1$(2a + b) - (a + b) = 0 - 1 \implies a = -1$ Substitute a = -1$a = -1$ back into equation (1): -1 + b = 1 implies b = 2$-1 + b = 1 \implies b = 2$ Substitute a = -1$a = -1$ into equation (2): -(-1) + c = 4 implies 1 + c = 4 implies c = 3$-(-1) + c = 4 \implies 1 + c = 4 \implies c = 3$ Thus, the explicit matrix A$A$ is: A = beginbmatrix 0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0 endbmatrix$A = \begin{\bmatrix} 0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0 \end{\bmatrix}$ ### Step 2: Compute Target Matrix Determinant Construct the modified target matrix 2(A+I)$2(A+I)$: A + I = beginbmatrix 1 & -1 & 2 \\ 1 & 1 & 3 \\ -2 & -3 & 1 endbmatrix$A + I = \begin{\bmatrix} 1 & -1 & 2 \\ 1 & 1 & 3 \\ -2 & -3 & 1 \end{\bmatrix}$ 2(A + I) = beginbmatrix 2 & -2 & 4 \\ 2 & 2 & 6 \\ -4 & -6 & 2 endbmatrix$2(A + I) = \begin{\bmatrix} 2 & -2 & 4 \\ 2 & 2 & 6 \\ -4 & -6 & 2 \end{\bmatrix}$ Calculate its determinant value: det(2(A+I)) = 2[4 - (-36)] - (-2)[4 - (-24)] + 4[-12 - (-8)]$\det(2(A+I)) = 2[4 - (-36)] - (-2)[4 - (-24)] + 4[-12 - (-8)]$ det(2(A+I)) = 2[40] + 2[28] + 4[-4] = 80 + 56 - 16 = 120$\det(2(A+I)) = 2[40] + 2[28] + 4[-4] = 80 + 56 - 16 = 120$ ### Step 3: Analyze Adjoint Power and Prime Factors Using the standard determinant identity for adjoints, det(mathrmadj(M)) = (det M)^n-1$\det(\mathrm{adj}(M)) = (\det M)^{n-1}$ where n=3$n=3$: det(mathrmadj(2(A+I))) = (120)^3-1 = 120^2$\det(\mathrm{adj}(2(A+I))) = (120)^{3-1} = 120^2$ Find the prime factorization of the result: 120 = 2^3 cdot 3^1 cdot 5^1 implies 120^2 = (2^3 cdot 3^1 cdot 5^1)^2 = 2^6 cdot 3^2 cdot 5^2$120 = 2^3 \cdot 3^1 \cdot 5^1 \implies 120^2 = (2^3 \cdot 3^1 \cdot 5^1)^2 = 2^6 \cdot 3^2 \cdot 5^2$ This maps the exponents directly to our target variables: alpha = 6, quad beta = 2, quad gamma = 2$\alpha = 6, \quad \beta = 2, \quad \gamma = 2$ Finally, calculate the \sum of their squares: alpha^2 + beta^2 + gamma^2 = 6^2 + 2^2 + 2^2 = 36 + 4 + 4 = 44$\alpha^2 + \beta^2 + \gamma^2 = 6^2 + 2^2 + 2^2 = 36 + 4 + 4 = 44$ ### Pattern Recognition The condition mathbfX^T A mathbfX = 0$\mathbf{X}^T A \mathbf{X} = 0$ always implies that A$A$ is a skew-symmetric matrix, which instantly forces the diagonal entries to be zero, reducing the number of unknown parameters from 9 down to 3. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants