If [tex]A=\left[\begin{array}{cc}4 & 2 \\ -7 & -3\end{array}\right][/tex] and [tex]B=\left[\begin{array}{cc}-3 & 4 \\ -1 & 5\end{array}\right][/tex], what is the solution to the equation [tex]A X=B[/tex]?

A. [tex]\left[\begin{array}{cc}-\frac{3}{4} & 2 \\ \frac{1}{7} & -1 \frac{2}{3}\end{array}\right][/tex]

B. [tex]\left[\begin{array}{cc}18 \frac{1}{2} & 11 \\ 19 & 11\end{array}\right][/tex]

C. [tex]\left[\begin{array}{cc}-14 & 26 \\ 24 & -43\end{array}\right][/tex]

D. [tex]\left\lceil 5 \frac{1}{2} \quad-11\right][/tex]



Answer :

To solve the equation \( A X = B \) where:

[tex]\[ A = \begin{pmatrix} 4 & 2 \\ -7 & -3 \end{pmatrix} \][/tex]

and

[tex]\[ B = \begin{pmatrix} -3 & 4 \\ -1 & 5 \end{pmatrix} \][/tex]

we need to find the matrix \( X \).

One way to solve this equation is by finding the inverse of matrix \( A \) and then multiplying it by matrix \( B \), because \( A^{-1} A X = A^{-1} B \), which simplifies to \( X = A^{-1} B \). However, we'll outline the result directly.

The solution to the equation \( A X = B \) gives us:

[tex]\[ X = \begin{pmatrix} 5.5 & -11 \\ -12.5 & 24 \end{pmatrix} \][/tex]

By inspecting the given options, this solution can be recognized as:

[tex]\[ \left\lceil 5 \frac{1}{2} \quad -11 \right] \][/tex]

Thus, the answer is:

[tex]\[ \left\lceil 5 \frac{1}{2} \quad -11 \right] \][/tex]

This matches the provided numerical result above. Note that the format slightly differs from standard matrix notation, but the numerical values match.

So the correct choice is:

[tex]\[ \left\lceil 5 \frac{1}{2} \quad -11 \right] \][/tex]