upāyaṁ karoti

Problem 23: For which value of $\lambda$ the given system has infinitely many solutions \begin{align*} 2x+3y+4z=13,\\ 5x+7y+7z=26,\\ 9x+13y+15z = 13\lambda. \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & b \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 2 & 3 & 4 & 13 \\ 5 & 7 & 7 & 26 \\ 9 & 13 & 15 & 13\lambda \\ \end{array}\right] \end{align*}\] \[\xrightarrow[R3\rightarrow R_3-R_2-2R_1]{R2\rightarrow R_2-2R_1} \left[\begin{array}{rrr|r} 2 & 3 & 4 & 13 \\ 1 & 1 & -1& 0 \\ 0 & 0 & 0 & 13\lambda -52 \\ \end{array}\right]\] for infinite solutions Rank[A]=Rank[A,b] = 2. So, \begin{align*} 13\lambda -52 =0 \Rightarrow \lambda = 4. \end{align*}

Problem 22: Investigate for what values a,b the equations \begin{align*} x+2y+3z=4,\\ x+3y+4z=5,\\ x+3y+az = b, \end{align*} have i. no solution, ii. a unique solution and iii. an infinite number of solutions. Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 1 & 3 & 4 & 5 \\ 1 & 3 & a & b \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-R_1,R3\rightarrow R_3-R_1} \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 0 & 1 & 1& 1 \\ 0 & 1 & a-3 & b-4 \\ \end{array}\right]\] \[\xrightarrow{R3\rightarrow R_3-R_2} \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 0 & 1 & 1& 1 \\ 0 & 0 & a-4 & b-5 \\ \end{array}\right]\] Case i. $a = 4$ and $b \ne 5$ \b...

Problem 21: Discuss for all values of $\lambda$, the system of equations \begin{align*} x+y+4z=6,\\ x+2y-2z=6,\\ \lambda x+y+z = 6, \end{align*} as regards existence and nature of solutions. Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 4 & 6 \\ 1 & 2 & -2& 6 \\ \lambda & 1 & 1 & 6 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-R_1,R3\rightarrow R_3-\lambda R_1} \left[\begin{array}{rrr|r} 1 & 1 & 4 & 6 \\ 0 & 1 & -6& 0 \\ 0 & 1-\lambda & 1-4\lambda & 6-6\lambda \\ \end{array}\right]\] Therefore the coefficient matrix will be non-singular if and only if, \begin{align*} 1-4\lambda + 6-6\lambda \ne 0,\\ \lambda \ne \frac{7}{10}. \end{align*} In Case $\lambda = \frac{7}{10}$, \begin{alig...

Problem 20: For what values of the $\eta$ the equations \begin{align*} x+y+z=1\\ x+2y+4z=\eta\\ x+4y+10z = {\eta}^2? \end{align*} have a solution and solve them completely in each case. Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1 & 1 \\ 1 & 2 & 4& \eta \\ 1 & 4 & 10 & {\eta}^2 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-R_1,R3\rightarrow R_3-R_1} \left[\begin{array}{rrr|r} 1 & 1 & 1 & 1 \\ 0 & 1 & 3& \eta-1 \\ 0 & 3 & 9 & {\eta}^2-1 \\ \end{array}\right]\] \[\xrightarrow{R3\rightarrow R_3-3R_2} \left[\begin{array}{rrr|r} 1 & 1 & 1 & 1 \\ 0 & 1 & 3& \eta-1 \\ 0 & 0 & 0 & {\eta}^2-3\eta +2 \\ \end{array}\right]\] Now the given equatio...

Problem 19: For what values of the $\lambda$ will the following equations fail to have unique solution \begin{align*} 3x-y+\lambda z =1\\ 2x+y+z=2\\ x+2y-\lambda z = -1? \end{align*} will the equations have any solution for these values of $\lambda$ Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 3 & -1 & \lambda & 1 \\ 2 & 1 & 1& 2 \\ 1 & 2 & -\lambda & -1 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R_1\leftrightarrow R_3 } \left[\begin{array}{rrr|r} 1 & 2 & -\lambda & -1 \\ 2 & 1 & 1& 2 \\ 3 & -1 & \lambda & 1 \\ \end{array}\right]\] \[\xrightarrow{R2\rightarrow R_2-2R_1,R3\rightarrow R_3-3R_1} \left[\begin{array}{rrr|r} 1 & 2 & -\lambda & -1 \\ 0 & -3 & 1+2\lambda & 4 \\ 0 ...

Problem 18: Investigate for what values of $\lambda, \mu$ the equations \begin{align*} x+y+z=6\\ x+2y+3z=10\\ x+2y+\lambda z = \mu \end{align*} have i. no solution, ii. a unique solution. iii. an infinite number of solutions. Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 1 & 2 & 3& 10 \\ 1 & 2 & \lambda& \mu \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-R_1,R3\rightarrow R_3-R_1 } \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 1 & \lambda-1& \mu-6 \\ \end{array}\right]\] \[\xrightarrow{R3\rightarrow R_3-R_2} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 0 & \lambda-3& \mu-10 \\ \end{array}\right]\] ...

Problem 17: State the conditions under which a system of non-homogeneous equations will have i. no solution ii. a unique solution iii. infinity of solutions. Solution:  Let $AX=B$ be a system of linear non-homogeneous equations, where $A,X,B$ are $m\times n,n\times 1,m\times 1$ matrices respectively.   i. These equations will have no solution if the coefficient matrix $A$ and augmented matrix $\left[A|B\right]$ are not of the same rank.   ii. These equations will possess a unique solution if the matrices $A$ and augmented matrix $\left[A|B\right]$ are of the same rank and the rank is equal to the number of variables.   iii. These equations will have infinity of solutions if the matrices $A$ and augmented matrix $\left[A|B\right]$ are of the same rank and the rank is less than the number of variables.

Problem 16: Discuss for all values of k the system of equations \begin{align*} 2x+3ky+(3k+4)z=0\\ x+(k+4)y+(4k+2)z=0\\ x+2(k+1)y+(3k+4)z=0 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 2 & 3k & 3k+4& 0 \\ 1 & k+4 & 4k+2& 0 \\ 1 & 2k+2 & 3k+4& 0 \\ \end{array}\right] \xrightarrow{R1\leftrightarrow R2} \left[\begin{array}{rrr|r} 1 & k+4 & 4k+2& 0 \\ 2 & 3k & 3k+4& 0 \\ 1 & 2k+2 & 3k+4& 0 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-2R_1,R3\rightarrow R_3-R_1 } \left[\begin{array}{rrr|r} 1 & k+4 & 4k+2& 0 \\ 0 & k-8 & -5k& 0 \\ 0 & k-2 & -k+2& 0 \\ \end{array}\right]\] \[\xrightarrow{R2\rightarrow \frac{1}{k-8}R_2} \left[\begin{array}...

Problem 15: For which values of k will the following system have i. No solution ? ii. Exactly one solution ? iii. Infinitely many solutions ? \begin{align*} 4x+y+(k^2-14)z=k+2\\ x+2y-3z=4\\ 3x-y+5z=2 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 4 & 1 & k^2-14& k+2 \\ 1 & 2 & -3& 4 \\ 3 & -1 & 5& 2 \\ \end{array}\right] \xrightarrow{R1\leftrightarrow R2} \left[\begin{array}{rrr|r} 1 & 2 & -3& 4 \\ 4 & 1 & k^2-14& k+2 \\ 3 & -1 & 5& 2 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\leftrightarrow R3} \left[\begin{array}{rrr|r} 1 & 2 & -3& 4 \\ 3 & -1 & 5& 2 \\ 4 & 1 & k^2-14& k+2 \\ \end{array}\right] \xrightarrow{R2\rightarrow R_2-3R_1,R3\rightar...

Problem 14: Determine whether the following system of equations are consistent or not. If yes, find all possible solutions: \begin{align*} x_1-2x_2-x_3+3x_4=0\\ -2x_1+4x_2+5x_3-5x_4=3\\ 3x_1-6x_2-6x_3+8x_4=2 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ -2 & 4 & 5& -5 & 3 \\ 3 & -6 & -6& 8 & 2 \\ \end{array}\right] \xrightarrow{R2\rightarrow R_2+2R_1,R3\rightarrow R_3-3R_1} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ 0 & 0 & 3& 1 & 3 \\ 0 & 0 & -3& -1 & 2 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R3\rightarrow R_3+R_2} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ 0 & 0 & 3& 1 & 3 \\ 0 & 0 &...

 Problem 13: Determine whether the following system of equations are consistent or not. If yes, determine all solutions: \begin{align*} x_1+x_2+2x_3=3\\ -x_1-3x_2+4x_3=2\\ -x_1-5x_2+10x_3=7 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ -1 & -3& 4 & 2 \\ -1 & -5 & 10 & 7 \\ \end{array}\right] \xrightarrow{R2\rightarrow R_2+R_1,R3\rightarrow R_3+R_1} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ 0 & -2& 6 & 5 \\ 0 & -4 & 12 & 10 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-2R_2} \left[\begin{array}{rrr|r} 1 & 1 & 2& 3 \\ 0 & -2& 6 & 5 \\ 0 & 0 & 0 & 0 \\ \end{array}\right] \xrightarrow{R_3\rightarrow -\frac{1}{2}...

 Problem 12: Determine whether the following system of equations are consistent or not. If yes, determine all solutions: \begin{align*} x_1+x_2+x_3=2\\ 2x_1+x_2-x_3=3\\ x_1+2x_2+4x_3=7 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & -1 & 1& 2 \\ 2 & 1& -1 & 3 \\ 1 & 2 & 4 & 7 \\ \end{array}\right] \xrightarrow{R2\rightarrow R_2-2R_1,R3\rightarrow R_3-R_1} \left[\begin{array}{rrr|r} 1 & -1 & 1& 2 \\ 0 & 3 & -3 & -1 \\ 0 & 3 & 3 & 5 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow \frac{1}{3}R_2,R3\rightarrow \frac{1}{3}R_3} \left[\begin{array}{rrr|r} 1 & -1 & 1& 2 \\ 0 & 1 & -1 & -\frac{1}{3} \\ 0 & 1 & 1 & \frac{5}{3} \\ \end{array}\righ...

 Problem 11: Solve the following systems of homogeneous equations by Gaussian elimination method: \begin{align*} x_1+x_2+x_3=0\\ x_1+2x_2+x_3=0\\ x_1+x_2+2x_3=0 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 0 \\ 1 & 2& 1 & 0 \\ 1 & 1 & 2 & 0 \\ \end{array}\right] \end{align*}\] \[\xrightarrow[R2\rightarrow R_2-R_1]{R3\rightarrow R_3-R_1} \left[\begin{array}{rrr|r} 1 & 1 & 1& 0 \\ 0 & 1& 1 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array}\right]\] For Gaussian elimination we write down the equations corresponding to REF. Then we get \begin{align*} x_1+x_2+x_3=0\\ x_2+x_3=0\\ x_3=0 \end{align*} By solving we get Solution set S is \begin{align*} x_1=x_2=x_3=0. \end{align*} The solution is trivial solut...

 Problem 10: Solve the following systems of linear equations by Gaussian elimination method: \begin{align*} 2x-2y+3z=2\\ x+2y-z=3\\ 3x-y+2z=1 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 2 & -2 & 3& 2 \\ 1 & 2& -1 & 3 \\ 3 & -1 & 2 & 1 \\ \end{array}\right] \xrightarrow{R_1\leftrightarrow R_2} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ 2 & -2 & 3& 2 \\ 3 & -1 & 2 & 1 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-2R_1,R3\rightarrow R_3-3R_1} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ 0 & -6 & 5& -4 \\ 0 & -7 & 5 & -8 \\ \end{array}\right] \xrightarrow{R3\rightarrow R_3-R_2} \left[\begin{array}{rrr|r} 1 & 2& -1 & 3 \\ ...

 Problem 9: Solve the following system by a. Gaussian elimination method b. Gauss-Jordan elimination method, c. Find particular solution of the system $AX = B$. \begin{align*} x_1+2x_2+x_3+x_4=0\\ 3x_1+4x_4=2\\ x_1-4x_2-2x_3-2x_4=2 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrrr|r} 1 & 2 & 1 & 1 & 0 \\ 3 & 0& 0 & 4 & 2 \\ 1 & -4 & -2 & -2 & 2 \\ \end{array}\right] \xrightarrow{R_2\rightarrow R_2-3R_1} \left[\begin{array}{rrrr|r} 1 & 2 & 1 & 1 & 0 \\ 0 & -6& -3 & 1 & 2 \\ 1 & -4 & -2 & -2 & 2 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-R_1} \left[\begin{array}{rrrr|r} 1 & 2 & 1 & 1 & 0 \\ 0 & -6& -3 & 1 & 2 \\ 0 ...

 Problem 8: Solve the following system by a. Gaussian elimination method b. Gauss-Jordan elimination method, c. Find particular solution of the system $AX = B$. \begin{align*} x_1+2x_2+x_3+x_4=1\\ 3x_1+4x_4=1\\ x_1-4x_2-2x_3-2x_4=0 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrrr|r} 1 & 2 & 1 & 1 & 1 \\ 3 & 0& 0 & 4 & 1 \\ 1 & -4 & -2 & -2 & 0 \\ \end{array}\right] \xrightarrow{R_2\rightarrow R_2-3R_1} \left[\begin{array}{rrrr|r} 1 & 2 & 1 & 1 & 1 \\ 0 & -6& -3 & 1 & -2 \\ 1 & -4 & -2 & -2 & 0 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-R_1} \left[\begin{array}{rrrr|r} 1 & 2 & 1 & 1 & 1 \\ 0 & -6& -3 & 1 & -2 \\ 0 ...

 Problem 7: Solve the following system of linear equations using Gaussian elimination. \begin{align*} x_1+2x_2+x_3=5\\ x_1+x_2+5x_3+2x_4=7\\ x_1+2x_2+8x_3+4x_4=12 \end{align*} Solution: Consider the augmented matrix of this system and apply row operations. \[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 1 & 1 & 5 & 2 & 7 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right] \xrightarrow{R_2\rightarrow R_2-R_1} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right]. \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-R_1} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 0 & 2 & 6 & 3 & 7 \\ \end{array}\right] \xrightarrow{R3\rightarrow R_3-...

 Problem 6: [Gaussian elimination] Let  \[\left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 &5 \\ 1 & 1 & 5 & 2 & 7 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right]\] be a given matrix. Obtain row-reduction of $A$ and reduced row reduction of $A$. Solution: We apply elementary row operations as follows to reduce the system to row echelon form. \[\begin{align*} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 1 & 1 & 5 & 2 & 7 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right] \xrightarrow{R_2\rightarrow R_2-R_1} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right]. \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-R_1} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 0 & 2 & 6 ...

Problem5: [ Homogeneous and Non-homogeneous System of Linear  Equations]   Which of the following set of equations are homogeneous.  1. $x_1+3x_2-x_3=0\\x_2-4x_3=0\\5x_3=0$ 2. $2x_1-3x_2=0\\4x_1-6x_2=0$ 3. $x_1+x_2+x_4=0\\x_2+x_3=0\\x_4=-1$ Solution: 1. Homogeneous - All right hand side constants are zero. 2. Homogeneous - All right hand side constants are zero. 3. Non-homogeneous - All right hand side constants are not zero. 

Quiz1: [ Systems of Linear Equations] - Linear Equations 1. State True or False: $\sqrt2x-\pi y+4\sqrt z= \sqrt5$ is a linear equation. True False 2. State True or False: $x^2+3y+z=6$ is a linear equation. True False 3. Which of the following equations are linear? a. $y=2x-5$ b. $y=x^2$ c. $x+2y=7$ d. $x^2+y^2=25$ e. $y=x^3$