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[tex]\[ A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad B=\begin{bmatrix} 22 & 11 & 7 \\ 6 & -9 & -15 \end{bmatrix}, \quad \text{and} \quad AB=\begin{bmatrix} 110 & -77 & -151 \\ 230 & 139 & 107 \end{bmatrix}. \][/tex]

Find the elements of matrix [tex]\( A \)[/tex].

[tex]\[ a = \boxed{} \][/tex]

[tex]\[ b = \boxed{} \][/tex]

[tex]\[ c = \boxed{} \][/tex]

[tex]\[ d = \boxed{} \][/tex]



Answer :

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]