PRE CALCULUS Matrices and Vectors 2

Question 1

What is the inverse of the matrix [[2, 1], [5, 3]]?

Accepted Answer

[[3, −1], [−5, 2]]

Answer

For [[a,b],[c,d]], the inverse is (1/det)[[d,−b],[−c,a]]; det = 1, so inverse = [[3,−1],[−5,2]].

Question 2

Which of the following represents the cross product result for 3D vectors?

Accepted Answer

A vector perpendicular to both

Answer

The cross product of two 3D vectors produces a vector perpendicular to both original vectors.

Question 3

How many solutions does a system have if its coefficient matrix has determinant = 0?

Accepted Answer

Zero or infinitely many

Answer

A zero determinant means the matrix is singular, so the system is either inconsistent (no solution) or dependent (infinitely many).

Question 4

What is the scalar projection of u = ⟨4, 3⟩ onto v = ⟨1, 0⟩?

Accepted Answer

4

Answer

The scalar projection of u onto v is (u · v)/|v| = (4·1 + 3·0)/1 = 4.

Question 5

A matrix A is called symmetric if:

Accepted Answer

A = Aᵀ

Answer

A symmetric matrix equals its own transpose, meaning element aᵢⱼ = aⱼᵢ for all i, j.

PRE CALCULUS Practice Test