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How many 4 x 4 matrices with entries from have odd determinant?
Hint: Use modulo arithmetic.
Hint: Use modulo arithmetic.
- a)20160
- b)32767
- c)49152
- d)53343
- e)65520
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
How many 4 x 4 matrices with entries from have odd determinant?Hint: U...
whenever 1st row is 0 then its determent is 0 , and similarly if any 2 or more rows are linearly dependent then its |det|=0 In order to find the odd determinant the
1st row must be non zero --> totally(2^4-1) possibilities |0/1 0/1 0/1 0/1| like totally=16-1
2nd row must be non zero and not linearly depends on 1st row so--> totally (2^4-2) possibilities
for 3rd row it must be non-zero as well as not linearly depends on first 2 rows(not start with 0) --->totally (2^4-4)
for 4th row -->(2^4-8) :: total possibilities=(2^4-2^0) * (2^4-2^1) * (2^4-2^2) *(2^4-2^3)=15*14*12*8=20160 possible
1st row must be non zero --> totally(2^4-1) possibilities |0/1 0/1 0/1 0/1| like totally=16-1
2nd row must be non zero and not linearly depends on 1st row so--> totally (2^4-2) possibilities
for 3rd row it must be non-zero as well as not linearly depends on first 2 rows(not start with 0) --->totally (2^4-4)
for 4th row -->(2^4-8) :: total possibilities=(2^4-2^0) * (2^4-2^1) * (2^4-2^2) *(2^4-2^3)=15*14*12*8=20160 possible
Most Upvoted Answer
How many 4 x 4 matrices with entries from have odd determinant?Hint: U...
Understanding the Problem
To find the number of 4 x 4 matrices with entries from the set of integers modulo 2 that have an odd determinant, we can leverage properties of linear algebra and modular arithmetic.
Key Points
- Matrix Entries: Each entry can be either 0 or 1, yielding 2 choices per entry.
- Total Matrices: The total number of 4 x 4 matrices is 2^(4*4) = 2^16 = 65536.
Determinant and Odd Result
- Determinant Modulo 2: A determinant is odd if it is equivalent to 1 modulo 2. In our case, this means we need to count how many matrices yield a determinant of 1 modulo 2.
Row Operations and Linear Independence
- Row Reduction: A 4 x 4 matrix is invertible if its rows are linearly independent.
- Counting Invertible Matrices: The number of invertible matrices over the field GF(2) can be calculated using the formula for the number of invertible matrices over a finite field.
Calculation Steps
1. First Row: Any non-zero vector (2^4 - 1 choices).
2. Second Row: Choose any vector not in the span of the first (2^4 - 2 choices).
3. Third Row: Choose any vector not in the span of the first two (2^4 - 4 choices).
4. Fourth Row: Choose any vector not in the span of the first three (2^4 - 8 choices).
The total number of invertible matrices is:
(2^4 - 1)(2^4 - 2)(2^4 - 4)(2^4 - 8) = (15)(14)(12)(8) = 20160.
Final Answer
Thus, the number of 4 x 4 matrices with entries from GF(2) that have an odd determinant is 20160 (option A).
To find the number of 4 x 4 matrices with entries from the set of integers modulo 2 that have an odd determinant, we can leverage properties of linear algebra and modular arithmetic.
Key Points
- Matrix Entries: Each entry can be either 0 or 1, yielding 2 choices per entry.
- Total Matrices: The total number of 4 x 4 matrices is 2^(4*4) = 2^16 = 65536.
Determinant and Odd Result
- Determinant Modulo 2: A determinant is odd if it is equivalent to 1 modulo 2. In our case, this means we need to count how many matrices yield a determinant of 1 modulo 2.
Row Operations and Linear Independence
- Row Reduction: A 4 x 4 matrix is invertible if its rows are linearly independent.
- Counting Invertible Matrices: The number of invertible matrices over the field GF(2) can be calculated using the formula for the number of invertible matrices over a finite field.
Calculation Steps
1. First Row: Any non-zero vector (2^4 - 1 choices).
2. Second Row: Choose any vector not in the span of the first (2^4 - 2 choices).
3. Third Row: Choose any vector not in the span of the first two (2^4 - 4 choices).
4. Fourth Row: Choose any vector not in the span of the first three (2^4 - 8 choices).
The total number of invertible matrices is:
(2^4 - 1)(2^4 - 2)(2^4 - 4)(2^4 - 8) = (15)(14)(12)(8) = 20160.
Final Answer
Thus, the number of 4 x 4 matrices with entries from GF(2) that have an odd determinant is 20160 (option A).
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