Let A and B be n-by-n matrices in R^n, and let c be a real number. Which of the following statements about trace is not necessarily true? a. tr(A+B) = tr(A) + tr(B) b. tr(AB) = tr(BA) c. tr(AB) = tr(A)tr(B) d. tr(CA) = ctr(A)

c. tr(AB) = tr(A)tr(B)

Step-by-step explanation:

The trace of a matrix is only valid for a square matrix, that is a n by n matrix. The trace of a matrix is the sum of all its diagonal elements. The following properties of trace holds for a matrix A and B with size n by n and a real number c.

i) The trace sum of two matrix is equal to the sum of their individual traces. That is:

tr(A + B) = tr(A) + tr(B)

ii) The trace of the product of a scalar and a matrix is the same as the product of the scalar and the trace of the product, that is:

tr(cA) = ctr(A)

iii) The trace of a transpose of a matrix is equal to the trace of the matrix, that is:

iv) The trace of a product of matrix is given as:

tr(AB) = tr(BA)

- 2 + 10x³ - 21x² + 19x - 4

step-by-step explanation:

each term in the second factor is multiplied by each term in the first factor

x²(- 2x² + 4x - 1) - 3x(- 2x² + 4x - 1) + 4(- 2x² + 4x - 1) ← distribute

= - 2 + 4x³ - x² + 6x³ - 12x² + 3x -8x² + 16x - 4

collect like terms

= - 2 + 10x³ - 21x² + 19x - 4 ← in standard form