1. Find the rank of the matrix [2 3; 1 3; 5]; 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... 1. Find the rank of the matrix [2 3; 1 3; 5]; 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 [3 7; 4 2; 3; 1 2 1] and show it satisfies the Cayley-Hamilton theorem; 4. If u = sin(x/y) + tan(y/x), prove that x du + y du = 0; 5. Verify Euler's theorem for am^2 + 2bxy + by^2; 6. If x = C(1+u), y = vC(1+u), find (partial du)/(partial xy); 7. If x = uV and y = uv - v, find (partial dC)/(partial xy); 8. If O(x,y,z) = 3m - y - y^3z^2, find nabla O at (1,-2,1); 9. Prove that div F = 3; 10. Evaluate the integral from 0 to a of (x^2+y^2) dx dy; 11. If F = x^2 i - 2x z j + 2y z k, find ∇F; 12. If u = e^(xyz), then prove that (partial^3 u)/(partial x partial y partial z) = (1 + 3yxz + x^2 y z^2)e^(xyz); 13. If z = (x^2 + y^2)/(x+y), then find (partial n)/(partial z); 14. If r = xi + yj + zk, find ∇r, where |r| = r and r = d(x^2 + y^2 + z^2); 15. If F = (x + y + 1) v i + j + (x - y) k, show that F' curl F = 0; 16. If r C(t) = 5t^2 i + tj + t^3 k, show that ∫(from 2) of (x (d^2 r)/(dt^2)) dt = -14 i + 75 j - 15 k; 17. If u = f(x/y), then find x ∂u/(∂x) + y ∂u/(∂y). Read more

Steps to Solve

1. Finding the Rank of the Matrix

To find the rank of the matrix $$\begin{bmatrix} 2 & 3 \ 1 & 5 \ 2 & 5 \end{bmatrix}$$, we perform row operations to reduce it to row echelon form.

2. Row Reduction Steps

Using elementary row operations:

• Subtract $\frac{1}{2}$ of the first row from the second row,
• Subtract the second row from the third row.

This gives us: $$\begin{bmatrix} 2 & 3 \ 0 & 1.5 \ 0 & -0.5 \end{bmatrix}$$

We can continue reducing this to find the number of non-zero rows.

3. Determining the Final Matrix Form

After reducing, the final form will show the leading 1s in each row.

4. Counting Non-Zero Rows

Count the number of rows that contain leading 1s. This number is the rank of the matrix.

5. Solving the Given System of Equations

We have the equations:

• $$2x + 3y - z = 0$$
• $$x - y - 2z = 0$$
• $$3x + ty + 3z = 0$$

Use substitution or elimination methods to solve these equations for $x$, $y$, and $z$.

6. Confirming Eigenvalues

For the matrix $$\begin{bmatrix} 3 & 7 \ 4 & 2 \ 1 & 1 \end{bmatrix}$$ find eigenvalues by solving the characteristic polynomial $$det(A - \lambda I) = 0$$ where $A$ is the given matrix.

7. Cayley-Hamilton Theorem

Once we have the eigenvalues, use them to verify that the matrix satisfies the Cayley-Hamilton theorem.

8. Verifying Using Calculus

For the functions and derivatives mentioned, apply the chain rule or product rule as needed and simplify.

9. Applying the gradient operator

Given $F = x^2 \hat{i} - 2xz \hat{j} + 2yz \hat{k}$, use the gradient operator to find $\nabla F$.