 # Linear Algebra Questions

## Transcribed Text

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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