1. (a) For the matrix compute the inverse, A-¹. A = 1 2. 0 03 -2 2 1 3 (b) Suppose you know that, for a matrix B, one has B-¹ = What is the solution of the following system of linear equations: Bx= 2 3 -2 1 1 1 ? 1 2 2 -1 2

For the following system of equations: X1 + 3x2 + x3 = -10 4x1 – 2x2 + 2x3 = -16 2x1 + x3 = -9 А. Find the PLU factorization of the coefficients matrix. Verify that PA = LU where L is a strict lower triangle matrix. Solve the system using the PLU factorization you found in part (a). В. С.

Translate the given system of equations into matrix form. x+y= z = 3 3x + y + z 4 3x 4 100 N +¾= 1 3 [-]

I. Consider the following linear system: 3x1 + 9x2 + 3x3 = 6 2x1 — 2х1 — 4x2 X3 = 12 = -8 We can view this system as a matrix equation: 3 9 31 0 -1 2 12 -2 -4 Which has the form: Ax X = B To solve the original system, we can solve this equation for X by multiplying by A-1: A-1x (A x X) = A-! × B (A-1 × A) × X = A-1 × B Iz x X = A-1 × B X = A-1 x B Use the bottom equation to solve for X. Find A-1 for the matrix A and solve the original system by using A-1 and Bin the (1) equation below. X = A-1 x B (2] Use the matrix A² you just found to solve the system: 3x1 + 9x2 + 3x3 = 12 2x1 -2x1 – 4x2 X3 = 9 = -3

3. Let A= a. 2 3 5 4 Find A¹ using the formula for the inverse of a 2x2 matrix. b. Write the following system of equations as a matrix equation of the form AX-B. Then solve the system using A-1¹. 2x+3y=-9 5x+4y=-5