Creating Matrix Equations for a System of Equations

Given Equations:

8x + 9y = 5

9x + 5y + 4z = -3

-3x - 3y + 10z = 5

(a) Make matrices for A and b such that the system of equations can be expressed as a matrix equation in the form A x = b.

(b) Is matrix A full rank?

Final answer:

(a) Matrices for the given system:

A = [[8, 9], [59, 5], [-3, -3], [-3, -3], [5, 10]]

b = [[9], [4z], [1], [z], [5]]

(b) Matrix A is not full rank in this case.

Explanation:

To express the system of equations in the form of a matrix equation A x = b, we need to arrange the coefficients in a matrix A and the constants on the right side of the equation in a matrix b.

Each row of matrix A corresponds to an equation and each column corresponds to a variable.

Here are the matrices for the given system:

A = [[8, 9], [59, 5], [-3, -3], [-3, -3], [5, 10]]

b = [[9], [4z], [1], [z], [5]]

For matrix A to be full rank, its rows or columns should be linearly independent. Here, the number of columns (2) is less than the number of rows (5), so matrix A is not full rank.