Mathematical Immaturity

2.20 Exercises

$\quad$ Apply the Gauss-Jordan elimination process to each of the following systems. If a solution exists, determine the general solution.

• \begin{align*} x + y + 3z &= 5 \\ 2x - y + 4z &= 11 \\ -y + x &= 3. \end{align*}
• \begin{align*} 3x + 2y + z &= 1 \\ 5x + 3y + 3z &= 2 \\ x + y - z &= 1. \end{align*}
• \begin{align*} 3x + 2y + z &= 1 \\ 5x + 3y + 3z &= 2 \\ 7x + 4y + 5z &= 3. \end{align*}
• \begin{align*} 3x + 2y + z &= 1 \\ 5x + 3y + 3z &= 2 \\ 7x + 4y + 5z &= 3 \\ x + y - z &= 0. \end{align*}
• \begin{align*} 3x - 2y + 5z + u &= 1 \\ x + y - 3z + 2u &= 2 \\ 6x + y - 4z + 3u &= 7. \end{align*}
• \begin{align*} x + y - 3z + u &= 5 \\ 2x - y + z - 2u &= 2 \\ 7x + y - 7z + 3u &= 3. \end{align*}
• \begin{align*} x + y + 2z + 3u + 4v &= 0 \\ 2x + 2y + 7z + 11u + 14v &= 0 \\ 3x + 3y + 6z + 10u + 15v &= 0. \end{align*}
• \begin{align*} x - 2y + z + 2u &= -2 \\ 2x + 3y - z - 5u &= 9 \\ 4x - y + z - u &= 5 \\ 5x - 3y + 2z + u &= 3. \end{align*}


• Prove that the system \begin{align*} x + y + 2z &= 2 \\ 2x - y + 3z &= 2 \\ 5x - y + az &= 6 \end{align*} has a unique solution if $a \neq 8.$ Find all solutions when $a = 8.$
• (a) $\quad$ Determine all the solutions of the system \begin{align*} 5x + 2y - 6z + 2u &= -1 \\ x - y + z - u &= -2 \end{align*} (b) $\quad$ Detemine all solutions of the system \begin{align*} 5x + 2y - 6z + 2u &= -1 \\ x - y + z - u &= -2 \\ x + y + z &= 6 \end{align*}
• This exercise tells how to determine all nonsingular $2 \times 2$ matrices. Prove that \begin{align*} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} &= (ad - bc)I. \end{align*} Deduce that $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ is nonsingular if and only if $ad - bc \neq 0,$ in which case its inverse is \begin{align*} \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}. \end{align*}

Determine the inverse of each of the matrices in Exercises 12 through 16.

• \begin{align*} \begin{bmatrix} 2 & 3 & 4 \\ 2 & 1 & 1 \\ -1 & 1 & 2 \end{bmatrix} \end{align*}
• \begin{align*} \begin{bmatrix} 1 & 2 & 2\\ 2 & -1 & 1 \\ 1 & 3 & 2 \end{bmatrix} \end{align*}
• \begin{align*} \begin{bmatrix} 1 & -2 & 1 \\ -2 & 5 & -4 \\ 1 & -4 & 6 \\ \end{bmatrix} \end{align*}
• \begin{align*} \begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \end{align*}
• \begin{align*} \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 & 0 & 1 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ \end{bmatrix} \end{align*}