If then the determinant of the matrix (A2016 – 2A2015

Engineering

Mathematics

Introduction to Matrix

Introduction to Determinants

Properties of Determinant

Question

If $A = [\begin{array}{cc} - 4 & - 1 \\ 3 & 1 \end{array}]$ then the determinant of the matrix (A²⁰¹⁶ – 2A²⁰¹⁵ – A²⁰¹⁴) is

– 175

– 25

– 2016

2016

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Given matrix A, we can factor the expression as A²⁰¹⁴(A² - 2A - I). First, compute A² - 2A - I.

Calculate A² = $[\begin{matrix} 13 & 3 \\ - 9 & - 2 \end{matrix}]$ , then A² - 2A - I = $[\begin{matrix} 13 + 8 - 1 & 3 + 2 - 0 \\ - 9 - 6 - 0 & - 2 - 2 - 1 \end{matrix}]$ = $[\begin{matrix} 20 & 5 \\ - 15 & - 5 \end{matrix}]$ = 5A.

So the expression becomes A²⁰¹⁴(5A) = 5A²⁰¹⁵. The determinant is |5A²⁰¹⁵| = 5²|A|²⁰¹⁵ = 25(-1)²⁰¹⁵ = 25(-1) = -25.

Final Answer: -25