Sure! Let's work through this problem step-by-step.
### Step 1: Define the matrix [tex]\( A \)[/tex]
Given:
[tex]\[ A = \left[\begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right] \][/tex]
### Step 2: Calculate [tex]\( A^2 \)[/tex]
First, we need to calculate [tex]\( A^2 \)[/tex] which is the matrix [tex]\( A \)[/tex] multiplied by itself:
[tex]\[ A^2 = A \times A \][/tex]
Using matrix multiplication:
[tex]\[ A = \left[\begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right] \][/tex]
[tex]\[ A^2 = \left[\begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right] \times \left[\begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right] \][/tex]
Calculating each element:
[tex]\[ (1,1) \text{ element: } (1 \cdot 1 + 2 \cdot 3) = 1 + 6 = 7 \][/tex]
[tex]\[ (1,2) \text{ element: } (1 \cdot 2 + 2 \cdot 1) = 2 + 2 = 4 \][/tex]
[tex]\[ (2,1) \text{ element: } (3 \cdot 1 + 1 \cdot 3) = 3 + 3 = 6 \][/tex]
[tex]\[ (2,2) \text{ element: } (3 \cdot 2 + 1 \cdot 1) = 6 + 1 = 7 \][/tex]
Thus:
[tex]\[ A^2 = \left[\begin{array}{cc} 7 & 4 \\ 6 & 7 \end{array} \right] \][/tex]
### Step 3: Calculate [tex]\( 2A \)[/tex]
Next, we calculate [tex]\( 2A \)[/tex] which is 2 multiplied by the matrix [tex]\( A \)[/tex]:
[tex]\[ 2A = 2 \times \left[\begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right] \][/tex]
[tex]\[ 2A = \left[\begin{array}{cc} 2 & 4 \\ 6 & 2 \end{array} \right] \][/tex]
### Step 4: Calculate [tex]\( 51I \)[/tex]
Here, [tex]\( I \)[/tex] is the identity matrix of the same size as [tex]\( A \)[/tex]:
[tex]\[ I = \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] \][/tex]
Thus, [tex]\( 51I \)[/tex] is:
[tex]\[ 51I = 51 \times \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] \][/tex]
[tex]\[ 51I = \left[\begin{array}{cc} 51 & 0 \\ 0 & 51 \end{array} \right] \][/tex]
### Step 5: Calculate the expression [tex]\( A^2 - 2A - 51I \)[/tex]
Now we combine our results to compute [tex]\( A^2 - 2A - 51I \)[/tex]:
[tex]\[ A^2 - 2A - 51I = \left[\begin{array}{cc} 7 & 4 \\ 6 & 7 \end{array} \right] - \left[\begin{array}{cc} 2 & 4 \\ 6 & 2 \end{array} \right] - \left[\begin{array}{cc} 51 & 0 \\ 0 & 51 \end{array} \right] \][/tex]
Subtracting element-wise:
[tex]\[ \left[\begin{array}{cc} 7-2-51 & 4-4-0 \\ 6-6-0 & 7-2-51 \end{array} \right] = \left[\begin{array}{cc} -46 & 0 \\ 0 & -46 \end{array} \right] \][/tex]
### Step 6: Calculate the determinant of the resulting matrix
Finally, we find the determinant:
[tex]\[ \text{Let } B = \left[\begin{array}{cc} -46 & 0 \\ 0 & -46 \end{array} \right] \][/tex]
The determinant of a diagonal matrix is the product of its diagonal elements:
[tex]\[ \det(B) = (-46) \times (-46) = 2116 \][/tex]
### Conclusion
The determinant of [tex]\( A^2 - 2A - 51I \)[/tex] is [tex]\( 2116 \)[/tex].
