Question

If A is $$ A = \begin{bmatrix} 8 & 7 \\5 & 6 \end{bmatrix} $$ then the value of $ \text{det}(A^{121} - A^{120}) = ? $

• A. 0
• B. 1
• C. 120
• D. 121

Hint:

Whenever the determinant of a matrix is multiplied by zero, the result will always be zero, regardless of the value of the other terms.

Answer 1

The Correct Option is A

First, let's understand the given expression \( \text{det}(A^{121} - A^{120}) \). Notice that: \[ A^{121} - A^{120} = A^{120}(A - I) \] where \( I \) is the identity matrix. Therefore, the determinant of the expression becomes: \[ \text{det}(A^{121} - A^{120}) = \text{det}(A^{120}) \times \text{det}(A - I) \] Next, consider that the matrix \( A \) has the following determinant: \[ \text{det}(A) = (8)(6) - (7)(5) = 48 - 35 = 13 \] Now, if we observe the structure of the matrix, it is easy to deduce that: \[ \text{det}(A^{120}) = (\text{det}(A))^{120} = 13^{120} \] So, we now need to compute \( \text{det}(A - I) \). We have: \[ A - I = \begin{bmatrix} 8-1 & 7 \\5 & 6-1 \end{bmatrix} = \begin{bmatrix} 7 & 7 \\5 & 5 \end{bmatrix} \] The determinant of \( A - I \) is: \[ \text{det}(A - I) = (7)(5) - (7)(5) = 35 - 35 = 0 \] Thus: \[ \text{det}(A^{121} - A^{120}) = 13^{120} \times 0 = 0 \] 
Conclusion: The value of \( \text{det}(A^{121} - A^{120}) \) is 0.