help!-matrix size is 3x3
clues for the encoding matrix:
1. all entries are one digit and positive
2. a11, a12, a13 are all equal and prime, their sum is also prime
3.the sum of a11, a12, a13 is equal to a21
4. the sum of a22 and the opposite of a33 is zero. a22 is prime and less than 7, greater than 3
5. a12 plus a22 equals a32
6. a21 minus a31 equals zero
7. a23 equals the sum of row 3 minus the sum of column 3
2. 1, 1, 1 sum = 3, no other sum is prime
3. 3
4. only prime between 3 and 7 is 5 so a22 is 5 so a33 is 5 since 5 -5 = 0
5. a12 + 5 = a32. If a12 = 1 then a32 = 6
6. a21=a31 = 3
7. a23 = 3 + 6 + 5 - 1 - 5 - a23
or 2 a23 = 8
so a23 = 4
so try
1 , 1 , 1
3 , 5 , 4
3 , 6 , 5
thanks
To find the encoding matrix, let's break down the given clues and solve them step by step:
1. All entries are one digit and positive: This tells us that each entry in the encoding matrix is a single-digit positive integer.
2. a11, a12, a13 are all equal and prime, their sum is also prime: Start by finding three prime numbers that are equal and whose sum is also prime. Prime numbers less than or equal to 9 are 2, 3, 5, and 7. Let's try each one:
- If a11 = 2, a12 = 2, a13 = 2, their sum is 6, which is not prime.
- If a11 = 3, a12 = 3, a13 = 3, their sum is 9, which is not prime.
- If a11 = 5, a12 = 5, a13 = 5, their sum is 15, which is not prime.
- If a11 = 7, a12 = 7, a13 = 7, their sum is 21, which is also not prime.
None of the options satisfy the given conditions, so let's proceed to the next clue.
3. The sum of a11, a12, a13 is equal to a21: Since clue 2 did not provide a valid solution, we need to compute the sum of a11, a12, and a13 and set it equal to a21. Let's consider the variables a11 = m, a12 = n, and a13 = o.
- From clue 3, we have m + n + o = 2m.
- Rearranging this equation, we get m = n + o.
Based on this equation, we can choose any two numbers as m and set the remaining variables equal to zero. For example, let's choose m = 2, n = 1, and o = 1.
So, a11 = 2, a12 = 1, a13 = 1, and a21 = 2 + 1 + 1 = 4.
4. The sum of a22 and the opposite of a33 is zero. a22 is prime and less than 7, greater than 3:
- We know that a22 is prime and less than 7. The prime numbers satisfying this condition are 5 and 3.
- We need to find a value for a33 such that a22 + (-a33) = 0. This means a33 is the negative of a22.
- Therefore, a22 = 5 and a33 = -5.
5. a12 plus a22 equals a32: Using the values from the previous clues, we can compute a32.
- a12 = 1, a22 = 5
- a32 = a12 + a22 = 1 + 5 = 6.
6. a21 minus a31 equals zero: With the values we already have, we can compute a31.
- a21 = 4
- a31 = a21.
7. a23 equals the sum of row 3 minus the sum of column 3:
- We know that:
* Sum of row 3 = a31 + a32 + a33 = a21 + a32 + a33 = 4 + 6 + (-5) = 5.
* Sum of column 3 = a13 + a23 + a33 = 1 + a23 + (-5) = -4 + a23.
- From clue 7, we have a23 = 5 - (-4 + a23).
* Simplifying, we get a23 = 5 + 4 - a23.
* Rearranging, we have 2a23 = 9.
* Dividing by 2, we get a23 = 4.5, but since all entries are one-digit positive integers, let's round it up to 5.
Now we have all the values for the encoding matrix:
a11 = 2, a12 = 1, a13 = 1
a21 = 4
a22 = 5, a33 = -5
a32 = 6
a31 = 4
a23 = 5
Therefore, the encoding matrix is:
| 2 1 1 |
| 4 0 -5 |
| 0 5 6 |
Please note that there might be alternate valid solutions based on different choices made during the process.