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Problem 1. Find out the solution set of the following system of linear equation. Represent the null space by the way you like(Try multible way is recommended) (1) 3x + 7y + 5z = 13 5x + 2y = 5 x + z = 3 (2) 3x + 3y + 5z = 11 x + 2y = 3 1 (3) x + y + z = 6 (4) 3x + 2y + z = 5 4x + y + 6z = 7 10x + 5y + 8z = 17 (5) 3x + 2y + z = 5 4x + y + 6z = 8 10x + 5y + 8z = 17 Problem 2. Find all possible value of λ for each of the following problem, (1) such that this linear Equation have unique solution (3 + λ)x + 2y + z = 5 4x + y + 6z = 8 10x + 5y + 8z = 17 (2) such that this lienar Equation have non-zero solution (1 + λ)x − 2y = 0 2x + (5 + λ)y = 0 Problem 3. Suppose you have two bottles of alcohol solution. The first bottle the concentration of alcohol is 30%, in the second bottle the concentration of alcohol is 50%, in which ratio will you mix them to get an alcohol solution have concentration 45%? 4 Problem 4. Now m, n is two fixed number, in a discussion titled ”which m × n matrix do you like”. The two people says” Ricky: I like all the matrix Am×n, such that the system of lienar equation Ax = b always has solution no matter what b is! Let’s call this kind of matrix as Ricky’s Matrix Michel: I like all the matrix Am×n, such that AT x = 0 only have 0 solution! Let’s call this kind of matrix as Michel’s Matrix Host: Well, I claim that every Ricky’s matrix is Michel’s matrix. and every Michel’s matrix is also Ricky’s matrix. Is that true? if it is, proof. (Hint: suppose y is a solution, consider 0 = 0x = (AT y) T x = y T Ax = y T b, bacause we can find x which could enable b to be any vector, that forces y to be 0) 1 Problem 1. What is the size and the 1-2 entry of the following matrix? 1 2 4 2 5 1 Problem 2. Which of the following are diagonal matrices? (1) 3 1 9 (2) 9 8 2 (3) 3 0 0 0 7 0 0 0 0 Problem 3. Compute the following matrix product. (1) 4 2 3 6 2 1 7 2 1 1 1 5 9 1 1 3 (2) 1 5 1 2 3 3 0 9 (3) 1 2 1 1 1 6 Problem 4. Find the inverse of the following matrix 1 1 2 1 6 1 1 1 2 1 1 Problem 5. Find a 2 × 2 matrices P, Q, such that P 1 2 5 9 Q = 1 1 Problem 6. Calculate the determinant: (1) 1 9 4 5 3 1 1 2 2 4 2 1 2 (2) 2 4 1 3 2 1 4 Problem 7. Find the rank of the matrix 1 5 9 1 1 1 3 7 11 Problem 8. Determine the real number λ, such that the following equation have a non-zero solutions. And with this value of λ, find out the solution of the system of equation. (λ − 5)x − 3y = 0 x + (λ − 1)y = 0 Problem 9. If A,B are n × n matrices, show that rank(AB) ≤ min{rank(A), rank(B)}

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Linear Algebra Questions
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