Question

I need to ask for clarification on the process of this question. It says:Solve each system using a matrix. 4x - 12y = -16x + 4y = 4I understand the concept of creating a matrix from a system. But, beyond that, I am unclear on the process of how to solve it. Thank you in advance for your help. 4 -12 -16 4 4

Answer 1

We have the following 2x2 system of equations:

`\begin{pmatrix}4 & -12 \\ 6 & 4\end{pmatrix}\cdot\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}-1 \\ 4\end{pmatrix}`

and we need the vector solution

`\begin{pmatrix}x \\ y\end{pmatrix}`

Then, we need to find the inverse matrix of the 2x2 matrix on the left hand side. For any 2x2 matrix A,

`A=\begin{pmatrix}a & b \\ c & d\end{pmatrix}`

the inverse is given as.

`A^(-1)=(1)/(\det A)\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}`

where detA denotes the determinant. Therefore, by means of the inverse matrix, the general solution for any 2x2 matrix will be

`\begin{pmatrix}x \\ y\end{pmatrix}=A^(-1)\begin{pmatrix}z \\ w\end{pmatrix}`

for any vector with entries z and w.

In our case, the determinat is

`\begin{gathered} \det \begin{pmatrix}4 & -12 \\ 6 & 4\end{pmatrix}=4*4-(6)(-12) \\ \det \begin{pmatrix}4 & -12 \\ 6 & 4\end{pmatrix}=16+72 \\ \det \begin{pmatrix}4 & -12 \\ 6 & 4\end{pmatrix}=88 \end{gathered}`

Therefore, the solution of our system will be

`\begin{gathered} \begin{pmatrix}x \\ y\end{pmatrix}=A^(-1)\begin{pmatrix}-1 \\ 4\end{pmatrix} \\ \text{with } \\ A^(-1)=(1)/(88)\begin{pmatrix}4 & 12 \\ -6 & 4\end{pmatrix} \end{gathered}`

Explicitly,

`\begin{pmatrix}x \\ y\end{pmatrix}=(1)/(88)\begin{pmatrix}4 & 12 \\ -6 & 4\end{pmatrix}\cdot\begin{pmatrix}-1 \\ 4\end{pmatrix}`

Now, lets make the product of the right hand side. It yields,

`\begin{pmatrix}x \\ y\end{pmatrix}=(1)/(88)\begin{pmatrix}4(-1)+(12)(4) \\ (-6)(-1)+(4)(4)\end{pmatrix}`

which gives

`\begin{gathered} \begin{pmatrix}x \\ y\end{pmatrix}=(1)/(88)\begin{pmatrix}-4+48 \\ 6+16\end{pmatrix} \\ \begin{pmatrix}x \\ y\end{pmatrix}=(1)/(88)\begin{pmatrix}44 \\ 22\end{pmatrix} \end{gathered}`

since 22x4=48 and 44x2=88, we have

`\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}(1)/(2) \\ (1)/(4)\end{pmatrix}`

Therefore, the solution of the system is

`\begin{gathered} x=(1)/(2)=0.5 \\ \text{and} \\ y=(1)/(4)=0.25 \end{gathered}`