Question

A. Convert this matrix equation into a system of equations. Explain why you decided to follow the steps you did.B. Once you have a system of equations, use Cramer’s rule to solve for x, y, and z. Explain

Answer 1

Given:

The matrix is

$\begin{bmatrix}{3} & {-6} & {1} \\ {-2} & {1} & {-10} \\ {4} & {10} & {-7}\end{bmatrix}\times\begin{bmatrix}{x} & {} & {} \\ {y} & {} & {} \\ {z} & {} & {}\end{bmatrix}=\begin{bmatrix}{25} & {} & {} \\ {-20} & {} & {} \\ {-11} & {} & {}\end{bmatrix}$

Find-:

(a)

Convert this matrix equation into a system

(b)

Solve the values x, y, and z.

Explanation-:

The multiplication of the matrix is

$\begin{gathered} =\begin{bmatrix}{3} & {-6} & {1} \\ {-2} & {1} & {-10} \\ {4} & {10} & {-7}\end{bmatrix}\times\begin{bmatrix}{x} & {} & {} \\ {y} & {} & {} \\ {z} & {} & {}\end{bmatrix} \\ \\ =\begin{bmatrix}{3x-6y+z} & {} & {} \\ {-2x+y-10z} & {} & {} \\ {4x+10y-7z} & {} & {}\end{bmatrix} \end{gathered}$

The matrix is equal

$\begin{bmatrix}{3x-6y+z} & {} & {} \\ {-2x+y-10z} & {} & {} \\ {4x+10y-7z} & {} & {}\end{bmatrix}=\begin{bmatrix}{25} & {} & {} \\ {-20} & {} & {} \\ {-11} & {} & {}\end{bmatrix}$

If two matrices are equal, then each element is equal

So, the system equation is:

$\begin{gathered} 3x-6y+z=25 \\ \\ -2x+y-10z=-20 \\ \\ 4x+10y-7z=-11 \end{gathered}$

(B)

The solve the equation

$\begin{gathered} 3x-6y+z=25.................(1) \\ \\ -2x+y-10z=-20..........(2) \\ \\ 4x+10y-7z=-11...........(3) \end{gathered}$

The value of x, y and z.

,

$\begin{gathered} 3x-6y-25=z \\ \\ then\text{ equation} \\ \\ -2x+y-10(3x-6y-25)=-20 \end{gathered}$$4x+10y-7(3x-6y-25)=-11$

The x, y and z.

$\begin{gathered} x=4 \\ \\ y=-2 \\ \\ z=1 \end{gathered}$