Answer :
Sure, let's solve the given mathematical expression step-by-step. The goal is to find the simplified form of the given matrix expression.
First, we need to calculate the numerator and denominator separately.
Step 1: Define matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
[tex]\[ A = \begin{pmatrix} 7 & -2 \\ 3 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} \][/tex]
We need to calculate [tex]\( \frac{-24 + 3B + \frac{1}{2}A - B}{A \cdot B} \)[/tex].
Step 2: Compute each component of the numerator
1. Calculate [tex]\( 3B \)[/tex]:
[tex]\[ 3B = 3 \cdot \begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} = \begin{pmatrix} -9 & 0 \\ -6 & 6 \end{pmatrix} \][/tex]
2. Calculate [tex]\( \frac{1}{2} A \)[/tex]:
[tex]\[ \frac{1}{2} A = \frac{1}{2} \cdot \begin{pmatrix} 7 & -2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} \frac{7}{2} & -1 \\ \frac{3}{2} & \frac{1}{2} \end{pmatrix} \][/tex]
3. Calculate [tex]\( -B \)[/tex]:
[tex]\[ -B = -\begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 2 & -2 \end{pmatrix} \][/tex]
4. Add all components of the numerator:
First, combine [tex]\( 3B \)[/tex] and [tex]\( -B \)[/tex]:
[tex]\[ 3B - B = \begin{pmatrix} -9 & 0 \\ -6 & 6 \end{pmatrix} + \begin{pmatrix} 3 & 0 \\ 2 & -2 \end{pmatrix} = \begin{pmatrix} -6 & 0 \\ -4 & 4 \end{pmatrix} \][/tex]
Next, add [tex]\( \frac{1}{2} A \)[/tex]:
[tex]\[ \begin{pmatrix} -6 & 0 \\ -4 & 4 \end{pmatrix} + \begin{pmatrix} \frac{7}{2} & -1 \\ \frac{3}{2} & \frac{1}{2} \end{pmatrix} = \begin{pmatrix} -6 + \frac{7}{2} & -1 \\ -4 + \frac{3}{2} & 4 + \frac{1}{2} \end{pmatrix} = \begin{pmatrix} -\frac{5}{2} & -1 \\ -\frac{5}{2} & \frac{9}{2} \end{pmatrix} \][/tex]
Finally, account for [tex]\( -24 \)[/tex]:
[tex]\[ \begin{pmatrix} -\frac{5}{2} & -1 \\ -\frac{5}{2} & \frac{9}{2} \end{pmatrix} - 24 = \begin{pmatrix} -\frac{5}{2} - 24 & -1 \\ -\frac{5}{2} - 24 & \frac{9}{2} - 24 \end{pmatrix} \][/tex]
This simplifies to:
[tex]\[ \begin{pmatrix} -\frac{53}{2} & -1 \\ -\frac{53}{2} & -\frac{39}{2} \end{pmatrix} \][/tex]
Step 3: Compute the denominator [tex]\( A \cdot B \)[/tex]
[tex]\[ A \cdot B = \begin{pmatrix} 7 & -2 \\ 3 & 1 \end{pmatrix} \cdot \begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} \][/tex]
Multiply these matrices:
[tex]\[ A \cdot B = \begin{pmatrix} 7 \cdot -3 + (-2) \cdot -2 & 7 \cdot 0 + (-2) \cdot 2 \\ 3 \cdot -3 + 1 \cdot -2 & 3 \cdot 0 + 1 \cdot 2 \end{pmatrix} = \begin{pmatrix} -21 + 4 & 0 - 4 \\ -9 - 2 & 0 + 2 \end{pmatrix} = \begin{pmatrix} -17 & -4 \\ -11 & 2 \end{pmatrix} \][/tex]
Step 4: Combine numerator and denominator
Now we have:
1. Numerator: [tex]\( \begin{pmatrix} -\frac{53}{2} & -1 \\ -\frac{53}{2} & -\frac{39}{2} \end{pmatrix} \)[/tex]
2. Denominator: [tex]\( \begin{pmatrix} -17 & -4 \\ -11 & 2 \end{pmatrix} \)[/tex]
Thus, the required fraction is:
[tex]\[ \frac{\begin{pmatrix} -\frac{53}{2} & -1 \\ -\frac{53}{2} & -\frac{39}{2} \end{pmatrix}}{\begin{pmatrix} -17 & -4 \\ -11 & 2 \end{pmatrix}} \][/tex]
This is a complex matrix division problem which would typically be handled by software or simplified further by a more advanced mathematical approach, but this is the detailed calculation up to the point we can elaborate manually.
First, we need to calculate the numerator and denominator separately.
Step 1: Define matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
[tex]\[ A = \begin{pmatrix} 7 & -2 \\ 3 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} \][/tex]
We need to calculate [tex]\( \frac{-24 + 3B + \frac{1}{2}A - B}{A \cdot B} \)[/tex].
Step 2: Compute each component of the numerator
1. Calculate [tex]\( 3B \)[/tex]:
[tex]\[ 3B = 3 \cdot \begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} = \begin{pmatrix} -9 & 0 \\ -6 & 6 \end{pmatrix} \][/tex]
2. Calculate [tex]\( \frac{1}{2} A \)[/tex]:
[tex]\[ \frac{1}{2} A = \frac{1}{2} \cdot \begin{pmatrix} 7 & -2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} \frac{7}{2} & -1 \\ \frac{3}{2} & \frac{1}{2} \end{pmatrix} \][/tex]
3. Calculate [tex]\( -B \)[/tex]:
[tex]\[ -B = -\begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 2 & -2 \end{pmatrix} \][/tex]
4. Add all components of the numerator:
First, combine [tex]\( 3B \)[/tex] and [tex]\( -B \)[/tex]:
[tex]\[ 3B - B = \begin{pmatrix} -9 & 0 \\ -6 & 6 \end{pmatrix} + \begin{pmatrix} 3 & 0 \\ 2 & -2 \end{pmatrix} = \begin{pmatrix} -6 & 0 \\ -4 & 4 \end{pmatrix} \][/tex]
Next, add [tex]\( \frac{1}{2} A \)[/tex]:
[tex]\[ \begin{pmatrix} -6 & 0 \\ -4 & 4 \end{pmatrix} + \begin{pmatrix} \frac{7}{2} & -1 \\ \frac{3}{2} & \frac{1}{2} \end{pmatrix} = \begin{pmatrix} -6 + \frac{7}{2} & -1 \\ -4 + \frac{3}{2} & 4 + \frac{1}{2} \end{pmatrix} = \begin{pmatrix} -\frac{5}{2} & -1 \\ -\frac{5}{2} & \frac{9}{2} \end{pmatrix} \][/tex]
Finally, account for [tex]\( -24 \)[/tex]:
[tex]\[ \begin{pmatrix} -\frac{5}{2} & -1 \\ -\frac{5}{2} & \frac{9}{2} \end{pmatrix} - 24 = \begin{pmatrix} -\frac{5}{2} - 24 & -1 \\ -\frac{5}{2} - 24 & \frac{9}{2} - 24 \end{pmatrix} \][/tex]
This simplifies to:
[tex]\[ \begin{pmatrix} -\frac{53}{2} & -1 \\ -\frac{53}{2} & -\frac{39}{2} \end{pmatrix} \][/tex]
Step 3: Compute the denominator [tex]\( A \cdot B \)[/tex]
[tex]\[ A \cdot B = \begin{pmatrix} 7 & -2 \\ 3 & 1 \end{pmatrix} \cdot \begin{pmatrix} -3 & 0 \\ -2 & 2 \end{pmatrix} \][/tex]
Multiply these matrices:
[tex]\[ A \cdot B = \begin{pmatrix} 7 \cdot -3 + (-2) \cdot -2 & 7 \cdot 0 + (-2) \cdot 2 \\ 3 \cdot -3 + 1 \cdot -2 & 3 \cdot 0 + 1 \cdot 2 \end{pmatrix} = \begin{pmatrix} -21 + 4 & 0 - 4 \\ -9 - 2 & 0 + 2 \end{pmatrix} = \begin{pmatrix} -17 & -4 \\ -11 & 2 \end{pmatrix} \][/tex]
Step 4: Combine numerator and denominator
Now we have:
1. Numerator: [tex]\( \begin{pmatrix} -\frac{53}{2} & -1 \\ -\frac{53}{2} & -\frac{39}{2} \end{pmatrix} \)[/tex]
2. Denominator: [tex]\( \begin{pmatrix} -17 & -4 \\ -11 & 2 \end{pmatrix} \)[/tex]
Thus, the required fraction is:
[tex]\[ \frac{\begin{pmatrix} -\frac{53}{2} & -1 \\ -\frac{53}{2} & -\frac{39}{2} \end{pmatrix}}{\begin{pmatrix} -17 & -4 \\ -11 & 2 \end{pmatrix}} \][/tex]
This is a complex matrix division problem which would typically be handled by software or simplified further by a more advanced mathematical approach, but this is the detailed calculation up to the point we can elaborate manually.
