1. _______ system of linear equation has no solution.

a. Inconsistent

b. Consistent

c. Dependent

d. Empty

Answer

Correct option is A) Inconsistent system of linear equation has no solution.

2. Determine whether the following system of linear equations have no solution, infinitely many solution or unique solutions.

x+2y=3,

2x+4y=15

a. No Solution

b. Infinitely Many Solution

c. Unique Solution

d. Cannot Be determined

Answer

a. No Solution

**Step 1:** Write down the given system of equations:

**Step 2:** Simplify Equation 2 by dividing both sides by 2:

Now, we have the system of equations:

**Step 3:** Subtract Equation 1 from Equation 3:

At this point, we have reached a contradiction. The equation 0=4.50=4.5 is not true, which means that there is no consistent solution to this system of equations.

Therefore, the system of equations has no solution. The correct answer is indeed option A) "No Solution."

3. Which of the following is NOT a property of the determinant of a matrix?

(a) The determinant of a triangular matrix is equal to the product of its diagonal elements. (b) The determinant of the transpose of a matrix is equal to the determinant of the matrix itself. (c) The determinant of a product of two matrices is equal to the product of the determinants of the two matrices. (d) The determinant of a matrix is equal to the determinant of its inverse.

**Answer**

**:** (d) The determinant of a matrix is equal to the determinant of its inverse.

**Explanation:** The determinant of a matrix is equal to the determinant of its inverse only if the matrix is non-singular.

4. What is the rank of the following matrix?

(a) 1 (b) 2 (c) 3 (d) None of the above

Answer

Answer: (c) 3

Explanation: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.

The given matrix has three linearly independent rows. Therefore, its rank is 3.

5. What is the eigenvalue of the following matrix corresponding to the eigenvector [1, 1, 1]?

(a) 3 (b) 6 (c) 9 (d) 12

Answer

(d) 12

Explanation: An eigenvalue of a matrix is a scalar value that satisfies the following equation:

where A is the matrix, x is the eigenvector, and lambda is the eigenvalue.

6. What is the diagonalization of the following matrix?

Answer

Answer: (b)

Explanation: The diagonalization of a matrix is a process of finding a matrix that is similar to the given matrix and has a diagonal form.

To diagonalize the given matrix, we can find its eigenvalues and eigenvectors. The eigenvalues of the matrix are 1 and 5, and the corresponding eigenvectors are [1, 1] and [-1, 1].

We can then construct the following matrix:

P = [[1, -1], [1, 1]] This matrix is invertible, and its inverse is given by:

P^-1 = [[1, 1], [-1, 1]] We can now diagonalize the given matrix as follows:

D = P^-1 * A * P D = [[1, 1], [-1, 1]] * [[1, 2], [3, 4]] * [[1, -1], [1, 1]] D = [[1, 1], [1, 1]] Therefore, the diagonalization of the given matrix is [[1, 1], [1, 1]].

7. What is the derivative of the following function with respect to "x"?

Answer

Explanation :

To find the derivative of the given function *f*(*x*), use the rules of differentiation and differentiate each term separately:

8. Consider the function *g*(*x*)=*x_3−3_x_2+2_x*+1. What is the x-coordinate of the local maximum of this function?

(a) 0

(b) 1

(c) 2

(d) 3

Answer

c. 2

Explanation

To find the x-coordinate of the local maximum of the function *g*(*x*), we need to find the critical points and then determine which one corresponds to a local maximum.

First, find the derivative of *g*(*x*) and set it equal to zero to find the critical points:

*g*′(*x*)=32−6+2

Now, solve for *x* when *g*′()=0:

3_x_2−6+2=0

We can solve this quadratic equation using the quadratic formula

we get the critical points . To determine which one corresponds to a local maximum, we can analyze the behavior of the function around these points. which is approximately 2.

So, the correct answer is option (c).

9. What conditions must be satisfied for a function to be considered continuous at a specific point?

a. The limit at the point should exist, but the function's value may differ. b. The limit at the point should not exist, but the function's value should be defined. c. The limit at the point should exist and match the function's value, and the function should be differentiable. d. The limit at the point should be undefined, and the function's value should be zero.

**Answer**

c. The limit at the point should exist and match the function's value, and the function should be differentiable.

**Explanation** Continuity at a specific point requires that three conditions are met: the value of the function at the point should exist, the limit of the function as it approaches the point should exist and match the function's value, and the function should be differentiable at that point.

10. Which of the following functions is not continuous at *x*=0?

a. *f*(*x*)=∣_x_∣ b. *g*(*x*)=1/x c. *h*(*x*)=*x^2+2x*+1 d. *k*(*x*)=sin(*x*)

**Answer**

b. *g*(*x*)=1/x

**Explanation 2:** The function *g*(*x*)=1/x is not continuous at =0 because it has a vertical asymptote at this point, leading to a discontinuity. The other functions listed are continuous at =0.

Module 2: Linear Algebra, Calculus and Optimization

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