F1.2 Question 40 on paper please **Linear Algebra: Systems of Equations and Matrices**---### 39. Understanding Solutions of Linear Systems**Problem Statement:**If the linear system\[ \begin{aligned} a_1x + b_1y + c_1z &= 0 \\ a_2x - b_2y + c_2z &= 0 \\ a_3x + b_3y - c_3z &= 0 \end{aligned}\]has only the trivial solution, what can be said about the solutions of the following system?\[ \begin{aligned} a_1x + b_1y + c_1z &= 3 \\ a_2x - b_2y + c_2z &= 7 \\ a_3x + b_3y - c_3z &= 11 \end{aligned}\]**Discussion:**The condition that the first system has only the trivial solution implies that the determinant of the coefficient matrix is non-zero, ensuring that the system is non-degenerate. This typically indicates that the second system has a unique solution considering the non-homogeneous nature of the system.---### 40. Maximum and Minimum Leading 1's in Reduced Row Echelon Form**Problem Statement:**- **a.** If \( A \) is a matrix with three rows and five columns, what is the maximum possible number of leading 1's in its reduced row echelon form?- **b.** If \( B \) is a matrix with three rows and six columns, what is the maximum possible number of parameters in the general solution of the linear system with augmented matrix \( B \)?- **c.** If \( C \) is a matrix with five rows and three columns, then what is the minimum possible number of rows of zeros in any row echelon form of \( C \)?**Discussion:**- **a.** The maximum number of leading 1's, which is synonymous with the rank, is the lesser of the number of rows or columns. Thus, for \( A \) (3 rows, 5 columns), the maximum number of leading 1's is 3.- **b.** For \( B \) (3 rows, 6 columns), the system can be expected to have \(6 - 3 = 3

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a₁x + b₁y+c₁z = 0 a₂x-b₂y + c₂z = 0 a3x + bzy-czz = 0 has only the trivial solution, what can be said about the solu- tions of the following system? a₁x + b₁y+c₁z = 3 a₂x-b₂y + c₂z = 7 a3x + b3y-C3z = 11 40. a. If A is a matrix with three rows and five columns, then what is the maximum possible number of leading I's in its reduced row echelon form? b. If B is a matrix with three rows and six columns, then what is the maximum possible number of parameters in the general solution of the linear system with augmented matrix B? c. If C is a matrix with five rows and three columns, then what is the minimum possible number of rows of zeros in any row echelon form of C? 41. Describe all possible reduced row echelon forms of a b c d]