Solve the system by solving the corresponding matrix equation with an inverse, if possible.

A)
B)
C)
E)

Question

Solve the system by solving the corresponding matrix equation with an inverse, if possible. ​  ​
​
A) ​ 
B) ​ 
C) ​ 
E) ​

Quizplus · Accepted Answer

We can write the system of equations as the matrix equation Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix:
\begin{bmatrix}1 & 1 \ 2 & 3\end{bmatrix}x = \begin{bmatrix}4 \ 10\end{bmatrix}
To solve for x, we can use the inverse matrix A^-1 to isolate x:
x = A^-1b
To find A^-1, we can use the formula:
A^-1 = (1/ad-bc) * \begin{bmatrix}d & -b \ -c & a\end{bmatrix}
where a, b, c, and d are the elements of A such that ad-bc ≠ 0.
In this case, a=1, b=1, c=2, and d=3, so we have:
A^-1 = (1/1(3)-1(2)) * \begin{bmatrix}3 & -1 \ -2 & 1\end{bmatrix} = \begin{bmatrix}-3 & 1 \ 2 & -1\end{bmatrix}
Now we can solve for x:
x = A^-1b = \begin{bmatrix}-3 & 1 \ 2 & -1\end{bmatrix} * \begin{bmatrix}4 \ 10\end{bmatrix} = \begin{bmatrix}-2 \ 4\end{bmatrix}
Therefore, the solution to the system is x = [-2, 4] and the best choice is A)

Solve the System by Solving the Corresponding Matrix Equation with an Inverse