To find the elements of matrix [tex]\( A \)[/tex] such that the matrix multiplication [tex]\( AB \)[/tex] results in the matrix given, we will set up a system of linear equations based on matrix multiplication rules.
Given:
[tex]\[
A = \left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right], \quad B = \left[\begin{array}{ccc}
22 & 11 & 7 \\
6 & -9 & -15
\end{array}\right], \quad AB = \left[\begin{array}{ccc}
110 & -77 & -151 \\
230 & 139 & 107
\end{array}\right]
\][/tex]
To find the elements of [tex]\( A \)[/tex], we need to solve for [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex].
Given the correct answer from the calculations, we have:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = 11 \][/tex]
[tex]\[ c = 11 \][/tex]
[tex]\[ d = -2 \][/tex]
Therefore, the elements of matrix [tex]\( A \)[/tex] are:
[tex]\[
a = \boxed{2}, \quad b = \boxed{11}, \quad c = \boxed{11}, \quad d = \boxed{-2}
\][/tex]
