Question

If \[ \left| \begin{pmatrix} 2017 & 2018 \\ 2019 & 2020 \end{pmatrix} \right| + \left| \begin{pmatrix} 2021 & 2022 \\ 2023 & 2024 \end{pmatrix} \right| = 2k \] find \( k^3 \).

Hint:

For determinants of \(2 \times 2\) matrices, use the formula: \[ \left| \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right| = ad - bc. \] Simplify each term and add the results carefully.

Answer 1

Step 1: Begin by calculating the determinants of the two matrices. The determinant of a \(2 \times 2\) matrix \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] is given by: \[ \text{Determinant} = ad - bc. \] For the first matrix \[ \begin{pmatrix} 2017 & 2018 \\ 2019 & 2020 \end{pmatrix}, \] the determinant is: \[ \left| \begin{pmatrix} 2017 & 2018 \\ 2019 & 2020 \end{pmatrix} \right| = (2017)(2020) - (2018)(2019). \] Simplifying: \[ 2017 \times 2020 = 4064340, \quad 2018 \times 2019 = 4064342, \] so the determinant is: \[ 4064340 - 4064342 = -2. \] For the second matrix \[ \begin{pmatrix} 2021 & 2022 \\ 2023 & 2024 \end{pmatrix}, \] the determinant is: \[ \left| \begin{pmatrix} 2021 & 2022 \\ 2023 & 2024 \end{pmatrix} \right| = (2021)(2024) - (2022)(2023). \] Simplifying: \[ 2021 \times 2024 = 4084644, \quad 2022 \times 2023 = 4084646, \] so the determinant is: \[ 4084644 - 4084646 = -2. \] Step 2: Now, we add the two determinants: \[ -2 + (-2) = -4. \] Step 3: According to the given equation \[ \left| \begin{pmatrix} 2017 & 2018 \\ 2019 & 2020 \end{pmatrix} \right| + \left| \begin{pmatrix} 2021 & 2022 \\ 2023 & 2024 \end{pmatrix} \right| = 2k, \] we have: \[ -4 = 2k \quad \Rightarrow \quad k = -2. \] Step 4: Finally, we calculate \( k^3 \): \[ k^3 = (-2)^3 = -8. \]