Linear algebra

What is the formula for finding the slope of a line?

Which of the following matrices is not invertible?

What is the rank of a matrix?

What is the solution to the system of equations: 2x + 3y = 8 4x - y = 10 (Single Answer: Score 6) A) x = 2, y = 1, B) x = 3, y = 2, C) x = 4, y = -2, D) x = 1, y = 3 Explanation: The correct answer is A) x = 2, y = 1. By solving the system of equations using elimination or substitution, we find that these values satisfy both equations. 5. Which of the following is a scalar quantity?

What is the determinant of the matrix [3 1; -2 4]?

The determinant of a 2x2 matrix [a b; c d] is given by ad - bc, so in this case, it is (3*4) - (1*(-2)) = 12 - (-2) = 10. 7. What is the eigenvector corresponding to the eigenvalue ? = 2 for the matrix [4 2; 1 3]?

Solving this system of equations gives us v = [1; 1]. 8. What is the dot product of vectors u = [2 3 -1] and v = [-4 1 5]?

The dot product of two vectors u = [u1 u2 u3] and v = [v1 v2 v3] is given by u1*v1 + u2*v2 + u3*v3, so in this case, it is (2*(-4)) + (3*1) + (-1*5) = -8 + 3 - 5 = -11. 9. Which of the following matrices represents a reflection about the y-axis?

What is the solution to the equation 3x - 5 = 7?

By isolating x on one side of the equation, we get 3x = 7 + 5, which simplifies to 3x = 12. Dividing both sides by 3 gives us x = 12/3. 11. Which of the following is a basis for R^3?