Please explain each step thoroughly using theorms and defini

Please explain each step thoroughly using theorms and definitions. Thanks.

Describe all solutions by the matrix equation A = 0 in parametric vector form: A = [1 3 -3 7 0 1 -4 5]

Solution

we have matrix

A =     1         3           -3          7

          0         1           -4          5

we can see that it has 2 rows and 4 columns so it is 2 X 4 matrix.

A x = 0

here \"x\" should be 4 x 1 in order to perform matrix multiplication

x = x1

      x2

      x3

      x4                        //since it has 4 x1 so it will have 4 variables

we reducd the A matrix

R1 -> R₁ - 3*R₂

A = 1     0      9      -8

       0      1     -4      5

A x = 0

1x₁ + 0x₂ +9x₃ -8x₄ = 0

0x₁ + 1x₂ -4x₃ +5x₄ = 0

x₁ +9x₃ -8x₄ = 0   =>   x₁ = -9x₃ +8x₄

x₂ - 4x₃ + 5x₄ = 0   =>   x₂ = 4x₃ -5x₄

x₃ and x₄ are free variables

                ( -9 )          ( 8 )

X =   x₃   ( 4)   + x₄    (-5)

                 (1)                (0)

                  (0)               (1)