1. Find the rank of the matrix [2 3; 1 3 5; 2 5 8]. 2. Solve the equations 2x + 3y - z = 0, x - y - 2z = 0, 3x + ty + 3z = 0 and find the values of x, y, and z. 3. Find the eigenv... 1. Find the rank of the matrix [2 3; 1 3 5; 2 5 8]. 2. Solve the equations 2x + 3y - z = 0, x - y - 2z = 0, 3x + ty + 3z = 0 and find the values of x, y, and z. 3. Find the eigenvalues of matrix [4 2 3; 1 2 1; 3 7]. Show that this matrix satisfies the Cayley Hamilton Theorem. 4. If u = sin(x/y) + tan(y/x), then prove that x(∂u/∂x) + y(∂u/∂y) = 0. 5. Verify Euler's Theorem for the function ax^2 + 2hxy + by^2. 6. If x = C(u + v), y = u(C + v), find ∂(C,u,v)/∂(x,y). 7. If x = uv and y = u + v, then find ∂(C,u,v)/∂(x,y). 8. If φ(x,y,z) = 3y - y^3z^2, then find ∇φ at (1,-2,1). 9. Prove that div ∇F = 3. 10. Evaluate ∫ from 0 to a of (x^2 + y^2) dx dy. 11. If F = x^2 i - 2x z j + 2y z k, then find ∇·F. 12. If u = e^(xyz), then prove that ∂^3u/∂x∂y∂z = (1 + 3xyz + x^2y^2z^2)e^(xyz). Read more

Steps to Solve

1. Finding the Rank of the Matrix

To find the rank of the matrix $$ A = \begin{bmatrix} 2 & 3 \ 1 & 5 \ 2 & 5 \end{bmatrix}, $$ we will perform row reduction to echelon form. The rank is the number of non-zero rows in its reduced form.

2. Solving the System of Equations

We need to solve the following system:

$$ \begin{aligned} 2x + 3y - z & = 0 \ x - y - 2z & = 0 \ 3x + ty + 3z & = 0 \end{aligned} $$

We can write this in matrix form as $$ \begin{bmatrix} 2 & 3 & -1 \ 1 & -1 & -2 \ 3 & t & 3 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}. $$ Then use row reduction methods.

3. Finding the Eigenvalues

For the matrix $$ B = \begin{bmatrix} 4 & 3 \ 2 & 1 \end{bmatrix}, $$ we will compute the characteristic polynomial using the determinant: $$ \text{det}(B - \lambda I) = 0 $$ where $I$ is the identity matrix and $\lambda$ are the eigenvalues.

4. Cayley-Hamilton Theorem

After finding the eigenvalues, we need to confirm that the matrix satisfies the Cayley-Hamilton theorem: $$ B^{n} + a_{n-1}B^{n-1} + \ldots + a_0I = 0. $$

5. Verifying Euler's Theorem

For the function $$ f(x,y,z) = ax^2 + 2bxy + by^2, $$ we will verify Euler's theorem which states: $$ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = 3f. $$ We will find each partial derivative and substitute them accordingly.

6. Finding the Divergence

Given the vector field $$ \mathbf{F} = x^2 \hat{i} - 2x \hat{j} + 2y \hat{k}, $$ the divergence is given by $$ \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}. $$

7. Proving the Identity

For the function $$ u = e^{xyz}, $$ we need to prove $$ \frac{\partial^3 u}{\partial x \partial y \partial z} = \left(1 + 3xyz + x^2y^2 - z^2\right)e^{xyz}. $$