If \(A_k=\left[\begin{array}{cc} k-1 & k \\ k-2 & k+1 \end{array}\right] \), then what is det(A1) + det(A2) + det(A3) + ... + det(A100) equal to ?
10000
Understanding the Problem: Summing Determinants of Matrices
The question asks us to find the sum of the determinants of a sequence of matrices \(A_k\), where \(k\) ranges from 1 to 100. The matrix \(A_k\) is given by:
\(A_k=\left[\begin{array}{cc} k-1 & k \\ k-2 & k+1 \end{array}\right]\)
We need to calculate det(\(A_1\)) + det(\(A_2\)) + det(\(A_3\)) + ... + det(\(A_{100}\)).
Calculating the Determinant of \(A_k\)
For a 2x2 matrix \(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\), the determinant is calculated as \(ad - bc\).
Applying this formula to the matrix \(A_k\):
\(\text{det}(A_k) = (k-1)(k+1) - (k)(k-2)\)
Let's expand and simplify the expression:
\(\text{det}(A_k) = (k^2 - 1) - (k^2 - 2k)\)
\(\text{det}(A_k) = k^2 - 1 - k^2 + 2k\)
\(\text{det}(A_k) = 2k - 1\)
So, the determinant of the matrix \(A_k\) is the expression \(2k - 1\).
Summing the Determinants
Now we need to find the sum of det(\(A_k\)) for \(k\) from 1 to 100. This can be written using summation notation as:
\(\text{Sum} = \sum_{k=1}^{100} \text{det}(A_k) = \sum_{k=1}^{100} (2k - 1)\)
This is the sum of the first 100 terms of the sequence \(2k-1\).
Let's look at the first few terms:
- For \(k=1\), det(\(A_1\)) = 2(1) - 1 = 1
- For \(k=2\), det(\(A_2\)) = 2(2) - 1 = 3
- For \(k=3\), det(\(A_3\)) = 2(3) - 1 = 5
- ...
- For \(k=100\), det(\(A_{100}\)) = 2(100) - 1 = 199
The sum is \(1 + 3 + 5 + ... + 199\). This is the sum of the first 100 odd numbers. The sum of the first \(n\) odd numbers is \(n^2\). In this case, \(n=100\), so the sum is \(100^2\).
\(\text{Sum} = 100^2 = 10000\)
Alternatively, we can use the properties of summation:
\(\sum_{k=1}^{100} (2k - 1) = \sum_{k=1}^{100} 2k - \sum_{k=1}^{100} 1\)
\(= 2 \sum_{k=1}^{100} k - \sum_{k=1}^{100} 1\)
The sum of the first 100 integers is given by the formula \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\). For \(n=100\):
\(\sum_{k=1}^{100} k = \frac{100(100+1)}{2} = \frac{100 \times 101}{2} = 50 \times 101 = 5050\)
The sum of 1 repeated 100 times is \(100 \times 1 = 100\).
Substituting these values back into the sum expression:
\(\text{Sum} = 2 \times (5050) - 100\)
\(\text{Sum} = 10100 - 100\)
\(\text{Sum} = 10000\)
Both methods yield the same result.
| k | \(A_k\) | det(\(A_k\)) = 2k - 1 |
|---|---|---|
| 1 | \(\left[\begin{array}{cc} 0 & 1 \\ -1 & 2 \end{array}\right]\) | 2(1) - 1 = 1 |
| 2 | \(\left[\begin{array}{cc} 1 & 2 \\ 0 & 3 \end{array}\right]\) | 2(2) - 1 = 3 |
| 3 | \(\left[\begin{array}{cc} 2 & 3 \\ 1 & 4 \end{array}\right]\) | 2(3) - 1 = 5 |
| ... | ... | ... |
| 100 | \(\left[\begin{array}{cc} 99 & 100 \\ 98 & 101 \end{array}\right]\) | 2(100) - 1 = 199 |
The sum of these determinants is 10000.
Revision Table: Key Concepts for Determinants and Summation
| Concept | Description | Formula/Rule |
|---|---|---|
| Determinant of 2x2 Matrix | A scalar value computed from the elements of a square matrix. | det\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\) = \(ad - bc\) |
| Summation Notation | Compact way to represent the sum of a sequence of terms. | \(\sum_{k=m}^{n} f(k) = f(m) + f(m+1) + ... + f(n)\) |
| Sum of First n Integers | The sum of the sequence 1, 2, 3, ..., n. | \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\) |
| Sum of First n Odd Numbers | The sum of the sequence 1, 3, 5, ..., (2n-1). | \(\sum_{k=1}^{n} (2k-1) = n^2\) |
Additional Information: Applications of Determinants and Summation
Determinants are fundamental in linear algebra and have many applications, including:
- Finding the inverse of a matrix.
- Solving systems of linear equations using Cramer's rule.
- Calculating the area of a triangle or volume of a tetrahedron defined by vectors.
- Determining if a matrix is invertible (a matrix is invertible if and only if its determinant is non-zero).
Summation, on the other hand, is a core concept in calculus, statistics, and discrete mathematics. It is used to:
- Define integrals as limits of sums.
- Calculate expected values and probabilities in statistics.
- Analyze series and sequences.
- Solve problems involving arithmetic and geometric progressions.
In this problem, we combined the concepts of determinants with summation to solve the problem efficiently. Recognizing the sequence of determinants (1, 3, 5, ..., 199) as the first 100 odd numbers provided a shortcut to the sum using the formula for the sum of odd numbers, but the general summation formula also confirmed the result.
Important Questions from Operations on Matrices
If \(A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\), then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?
If \(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]\), then which of the following statements are correct?
1. An will always be singular for any positive integer n.
2. An will always be a diagonal matrix for any positive integer n.
3. An will always be a symmetric matrix for any positive integer n.
Select the correct answer using the code given below:
If \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then what is 23A3 - 19A2 - 4A equal to ?
Consider the following in respect of the matrix \({\rm{A}} = \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right):\)
1. A 2= -A
2. A 3= 4A
Which of the above is/are correct?If A is an orthogonal matrix of order 3 and \({\rm{B}} = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ { - 3}&0&2\\ 2&5&0 \end{array}} \right]\) , then which of the following is/are correct?
1. |AB| = ± 47
2. AB = BA
Select the correct answer using the code given below:
