Write down the following equations in matrix form. Using inverse of matrix, solve the equations, X + Y + Z = 6 3X – Y + 3Z = 10 5X + 5Y – 4Z = 3 

3 Answers | Add Yours

embizze's profile pic

Posted on

The instructions ask us to solve the system using inverse matrices:

 

The matrix form of the equation is:

`([1,1,1],[3,-1,3],[5,5,-4])([x],[y],[z])=([6],[10],[3])`

To solve an equation of the form `AX=B` we find the inverse of A, then `A^(-1)AX=A^(-1)B ==> X=A^(-1)B` making sure to multiply in the same order.

Using technology the inverse is `([-11/36,1/4,1/9],[3/4,-1/4,0],[5/9,0,-1/9])`

Then `X=([x],[y],[z])=([-11/36,1/4,1/9],[3/4,-1/4,0],[5/9,0,-1/9])([6],[10],[3])`

or `X=([1],[2],[3])`

----------------------------------------------------------------

If you are asked to find the inverse by hand then we use Gauss-Jordan elimination on the augmented matrix:

`([1,1,1:1,0,0],[3,-1,3:0,1,0],[5,5,-4:0,0,1])` until the left side is in reduced row echelon form.

sciencesolve's profile pic

Posted on

You need to collect the coefficients of variables x,y,z to form the matrix of the system, such that:

A = ((1,1,1),(3,-1,3),(5,5,-4))

You may evaluate the determinant of matrox of system, such that:

`Delta = [(1,1,1),(3,-1,3),(5,5,-4)] ` => `Delta = 4 + 15 + 15 + 5 - `

`15 + 12 Delta = 36`

Since `Delta != 0` , you may solve the system of equations using Cramer's method, such that:

`x = (Delta_x)/Delta, y = (Delta_y)/Delta, z = (Delta_z)/Delta`

`Delta_x = [(6,1,1),(10,-1,3),(3,5,-4)]`

`Delta_x = 24 + 50 + 9 + 3 - 90 + 40 = 36`

`x = 36/36 => x = 1`

`Delta_y = [(1,6,1),(3,10,3),(5,3,-4)] ` => `Delta_y = -40 + 9 + 90 - 50 - 9 + 72 = 72`

`y = 72/36 => y = 2`

Replacing 1 for x and 2 for y in the top equation `x + y + z = 6 ` yields:

`1 + 2 + z = 6 => z = 3`

Hence, evaluating the solution to the system of equations, using matrices, yields` x = 1, y = 2, z = 3.`

devdeepguha's profile pic

Posted on

Thank you soo much

We’ve answered 320,195 questions. We can answer yours, too.

Ask a question