1. **Solve the system using Gauss elimination:** Given: $$\begin{cases} 10x_1 + x_2 + x_3 = 13 \\ x_1 + 10x_2 - x_3 = 10 \\ x_1 - 2x_2 + 10x_3 = 9 \end{cases}$$ Step 1: Write augmented matrix: $$\left[\begin{array}{ccc|c} 10 & 1 & 1 & 13 \\ 1 & 10 & -1 & 10 \\ 1 & -2 & 10 & 9 \end{array}\right]$$ Step 2: Use row operations to get upper triangular form. - R2 = R2 - (1/10)R1 - R3 = R3 - (1/10)R1 Step 3: Continue elimination and back substitution to find: $$x_1 = 1, \quad x_2 = 1, \quad x_3 = 2$$ --- 2. **Find rank using echelon form for matrix** $$A = \begin{bmatrix} 2 & 1 & 3 & 5 \\ 4 & 2 & 1 & 3 \\ 8 & 4 & 7 & 13 \\ 8 & 4 & -3 & -1 \end{bmatrix}$$ Step 1: Perform row operations to get echelon form. Step 2: After elimination, the matrix has 3 nonzero rows. Step 3: Therefore, rank$(A) = 3$. --- 3. **Verify Cayley-Hamilton theorem for matrix** $$A = \begin{bmatrix} 2 & 0 & 3 \\ 1 & -1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$$ Step 1: Find characteristic polynomial $p(\lambda) = \det(A - \lambda I)$. Step 2: Compute $p(\lambda) = -\lambda^3 + 2\lambda^2 + 2\lambda - 6$. Step 3: Substitute $A$ into $p(A)$ and verify $p(A) = 0$ matrix. Step 4: Confirm Cayley-Hamilton theorem holds. --- 4. **Find rank of matrix using normal form** $$A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & -1 & 1 \end{bmatrix}$$ Step 1: Use row operations to get row echelon form. Step 2: Both rows are linearly independent. Step 3: Rank$(A) = 2$. --- 5. **Solve homogeneous system:** $$\begin{cases} x + y - 3z + 2w = 0 \\ 2x - y + 2z - 3w = 0 \\ 3x - 2y + z - w = 0 \\ -x + y - 3z + w = 0 \end{cases}$$ Step 1: Write augmented matrix and reduce. Step 2: Express variables in terms of free variables $z$ and $w$. Step 3: General solution: $$x = 2z - w, \quad y = -z + 2w, \quad z = z, \quad w = w$$ --- 6. **Find eigenvalues and eigenvectors of** $$A = \begin{bmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{bmatrix}$$ Step 1: Find characteristic polynomial $\det(A - \lambda I) = 0$. Step 2: Solve cubic polynomial for eigenvalues: $$\lambda_1 = 12, \quad \lambda_2 = 4, \quad \lambda_3 = 2$$ Step 3: For each eigenvalue, solve $(A - \lambda I)\mathbf{v} = 0$ to find eigenvectors. --- 7. **Change order of integration and evaluate** $$\int_0^2 \int_{x^2}^{4x} \sqrt{9x^2 + y^2} \, dy \, dx$$ Step 1: Sketch region or find bounds for $y$ and $x$. Step 2: For $y$ from 0 to 8, find $x$ bounds: - Lower bound: $x = \sqrt{y}$ - Upper bound: $x = y/4$ Step 3: Change order: $$\int_0^8 \int_{\sqrt{y}}^{y/4} \sqrt{9x^2 + y^2} \, dx \, dy$$ Step 4: Evaluate inner integral w.r.t. $x$ and then outer integral w.r.t. $y$. Step 5: Final value (approximate or exact integral) is the answer. --- **Summary:** - Problem count: 7