Question

If A_n = \(\begin{bmatrix} 1-n & n \\ n & 1-n \\ \end{bmatrix}\) then | A₁ | + | A₂ | + ….. + | A₂₀₂₁ | =

• A. -2021
• B. (2021)²
• C. -(2021)²
• D. 4042

Answer 1

The Correct Option is C

We are given: \[ A_n = \begin{bmatrix} 1 - n & n \\ n & 1 - n \end{bmatrix} \] Step 1: Determinant of \(A_n\) Use the formula for 2×2 determinant: \[ |A_n| = (1 - n)(1 - n) - (n)(n) = (1 - n)^2 - n^2 \] Simplify: \[ |A_n| = 1 - 2n + n^2 - n^2 = 1 - 2n \] Step 2: Sum of determinants from \(n = 1\) to \(2021\) \[ \sum_{n=1}^{2021} |A_n| = \sum_{n=1}^{2021} (1 - 2n) = \sum_{n=1}^{2021} 1 - \sum_{n=1}^{2021} 2n \] First sum: \[ \sum_{n=1}^{2021} 1 = 2021 \] Second sum: \[ \sum_{n=1}^{2021} 2n = 2 \cdot \frac{2021 \cdot (2021 + 1)}{2} = 2021 \cdot 2022 \] Now compute: \[ \sum_{n=1}^{2021} |A_n| = 2021 - 2021 \cdot 2022 = 2021(1 - 2022) = 2021 \cdot (-2021) = \mathbf{-(2021)^2} \]

Answer 2

We are given the matrix \(A_n = \begin{bmatrix} 1-n & n \\ n & 1-n \end{bmatrix}\) and we need to find the sum of the determinants |A₁| + |A₂| + ... + |A₂₀₂₁|.

First, let's find the determinant of A_n:

\(|A_n| = (1-n)(1-n) - (n)(n) = (1 - 2n + n^2) - n^2 = 1 - 2n\)

Now we want to find the sum:

S = |A₁| + |A₂| + ... + |A₂₀₂₁|

S = (1 - 2(1)) + (1 - 2(2)) + ... + (1 - 2(2021))

S = \(\sum_{n=1}^{2021} (1 - 2n)\)

We can split the summation:

S = \(\sum_{n=1}^{2021} 1 - 2 \sum_{n=1}^{2021} n\)

We know that \(\sum_{n=1}^{N} 1 = N\) and \(\sum_{n=1}^{N} n = \frac{N(N+1)}{2}\).

In this case, N = 2021.

So, \(\sum_{n=1}^{2021} 1 = 2021\)

And, \(\sum_{n=1}^{2021} n = \frac{2021(2021+1)}{2} = \frac{2021 * 2022}{2} = 2021 * 1011\)

Therefore,

S = 2021 - 2 * (2021 * 1011)

S = 2021 - 2 * 2043231

S = 2021 - 4086462

S = -4084441

The result -4084441 do not show in option lets rechek it the determinant is \(|A_n| = (1-n)(1-n) - (n)(n) = (1 - 2n + n^2) - n^2 = 1 - 2n\).

So the \(\sum_{n=1}^{2021} (1 - 2n)\) = ( \(\sum_{n=1}^{2021} 1\)) -(2\(\sum_{n=1}^{2021} n\) ) = 2021 - 2 * ( 2021 * 2022 / 2) =2021 - 2021 * 2022 = 2021( 1 - 2022) = 2021(-2021) = - (2021)²

S = -(2021)²

Answer: -(2021)²