Finding the Maximum Determinant Value with Prime Numbers
The question asks for the maximum value of a 2x2 determinant \(A = \left| {\begin{array}{*{20}{c}} p&q\\ r&s \end{array}} \right|\), where p, q, r, and s are four different prime numbers less than 20.
First, let's list the prime numbers less than 20. These are numbers greater than 1 that have no positive divisors other than 1 and themselves.
2
3
5
7
11
13
17
19
There are 8 prime numbers less than 20. We need to choose four different numbers from this list for p, q, r, and s.
The formula for the determinant of a 2x2 matrix \(\left| {\begin{array}{*{20}{c}} p&q\\ r&s \end{array}} \right|\) is:
\[ \text{Determinant} = ps - qr \]
To maximize the value of the determinant \(ps - qr\), we need to make the product \(ps\) as large as possible and the product \(qr\) as small as possible. Since p, q, r, and s must be four different prime numbers from the list {2, 3, 5, 7, 11, 13, 17, 19}, we should choose the numbers for p, s, q, and r accordingly.
To maximize \(ps\), we should select the two largest prime numbers from the list for p and s. The two largest prime numbers less than 20 are 19 and 17. So, {p, s} should be {19, 17}.
To minimize \(qr\), we should select the two smallest prime numbers from the *remaining* list for q and r. After choosing 19 and 17 for p and s, the remaining prime numbers are {2, 3, 5, 7, 11, 13}. The two smallest prime numbers from this remaining list are 2 and 3. So, {q, r} should be {2, 3}.
Now we can calculate the maximum determinant value using these chosen numbers. We can assign the values in different ways:
Case 1: p=19, s=17, q=2, r=3
Case 2: p=19, s=17, q=3, r=2
Case 3: p=17, s=19, q=2, r=3
Case 4: p=17, s=19, q=3, r=2
Let's calculate the determinant for one of these cases, for example, Case 1:
As you can see, the specific assignment of 19 and 17 to p or s, and 2 and 3 to q or r, does not change the final result because multiplication is commutative (\(19 \times 17 = 17 \times 19\)) and the overall structure is \(ps - qr\). The maximum value is consistently 317.
Chosen Numbers
p, s (Largest Two)
q, r (Smallest Two Remaining)
Determinant \(ps - qr\)
Prime numbers < 20: {2, 3, 5, 7, 11, 13, 17, 19}
19, 17
2, 3
\((19 \times 17) - (2 \times 3) = 323 - 6 = 317\)
Thus, the maximum value of the determinant is 317.
Revision Table: Key Concepts
Concept
Description
Relevance to Problem
Prime Number
A natural number greater than 1 that has no positive divisors other than 1 and itself.
The elements of the determinant must be four different prime numbers less than 20.
Determinant of a 2x2 Matrix
For \(\left| {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right|\), the determinant is \(ad - bc\).
We needed to calculate and maximize this value for the given matrix A.
Maximizing an Expression \(X - Y\)
To maximize \(X - Y\), maximize X and minimize Y.
Applied to maximize \(ps - qr\) by maximizing \(ps\) and minimizing \(qr\).
Additional Information: Determinants and Prime Numbers
Determinants are fundamental in linear algebra. They are scalar values calculated from the elements of a square matrix. They have various applications, such as determining if a matrix is invertible (a non-zero determinant means the matrix is invertible), finding the area of a parallelogram or volume of a parallelepiped defined by vectors, and solving systems of linear equations using Cramer's rule.
Prime numbers are the building blocks of integers through multiplication (Fundamental Theorem of Arithmetic). Their distribution is a major area of study in number theory. The concept of prime numbers is simple, but their properties and patterns are complex and have fascinated mathematicians for centuries. Using prime numbers in problems often leads to unique constraints and interesting mathematical structures, as seen in this problem where selecting specific prime numbers leads to the maximum determinant value.
This problem combines concepts from both number theory (prime numbers) and linear algebra (determinants), requiring an understanding of both definitions and properties to arrive at the solution.
Was this answer helpful?
Similar Questions
The value of the determinant \(\left| {\begin{array}{*{20}{c}} {1 - {\rm{\alpha }}}&{{\rm{\alpha }} - {{\rm{\alpha }}^2}}&{{{\rm{\alpha }}^2}}\\ {1 - {\rm{\beta }}}&{{\rm{\beta }} - {{\rm{\beta }}^2}}&{{{\rm{\beta }}^2}}\\ {1 - {\rm{\gamma }}}&{{\rm{\gamma }} - {{\rm{\gamma }}^2}}&{{{\rm{\gamma }}^2}} \end{array}} \right|\) is equal to
Let p, q and r be three distinct positive real numbers. If \(\rm D = \left| {\begin{array}{*{20}{c}} \rm p&\rm q&\rm r\\ \rm q&\rm r&\rm p\\ \rm r&\rm p&\rm q \end{array}} \right|,\) then which one of the following is correct?
What is the value of the determinant \(\left| {\begin{array}{*{20}{c}} {\rm{i}}&{{{\rm{i}}^2}}&{{{\rm{i}}^3}}\\ {{{\rm{i}}^4}}&{{{\rm{i}}^6}}&{{{\rm{i}}^8}}\\ {{{\rm{i}}^9}}&{{{\rm{i}}^{12}}}&{{{\rm{i}}^{15}}} \end{array}} \right|\) where \(\rm i = \sqrt {-1}\) ?
If \(\left| {\begin{array}{*{20}{c}} {\rm{x}}&{\rm{y}}&0\\ 0&{\rm{x}}&{\rm{y}}\\ {\rm{y}}&0&{\rm{x}} \end{array}} \right| = 0\) , then which one of the following is correct?
If x, y, z are distinct real numbers and \(\left| {\begin{array}{*{20}{c}} x&{{x^2}}&{2 + {x^3}}\\ y&{{y^2}}&{2 + {y^3}}\\ z&{{z^2}}&{2 + {z^3}} \end{array}} \right| = 0\), then xyz =
If A + B + C = \(\pi \), then, the value of \(\left| {\begin{array}{*{20}{c}} {\sin \left( {A + B + C} \right)}&{\sin B}&{\cos C}\\ { - \sin B}&0&{\tan A}\\ {\cos \left( {A + B} \right)}&{ - \tan A}&0 \end{array}} \right|\) is
Let p, q and r be three distinct positive real numbers. If \(\rm D = \left| {\begin{array}{*{20}{c}} \rm p&\rm q&\rm r\\ \rm q&\rm r&\rm p\\ \rm r&\rm p&\rm q \end{array}} \right|,\) then which one of the following is correct?
