Question
Show that if $a d-b c \neq 0$, then the system\[\begin{array}{l}a x+b y=r \\c x+d y=s\end{array}\]has a unique solution.
Step 1
The system of equations is \[ \begin{array}{l} a x+b y=r \\ c x+d y=s \end{array} \] which can be written in matrix form as \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} r \\ s \end{bmatrix} \] Show more…
Show all steps
Your feedback will help us improve your experience
Sikandar Baig and 83 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
If . $A=\left[\begin{array}{lll}a & 0 & 1 \\ 1 & c & b \\ 1 & d & b\end{array}\right], B=\left[\begin{array}{lll}a & 1 & 1 \\ 0 & d & c \\ f & g & h\end{array}\right], U=\left[\begin{array}{l}f \\ g \\ h\end{array}\right], V=\left[\begin{array}{c}a^{2} \\ 0 \\ 0\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $A X=U$ has infinitely many solutions, prove that $B X=V$ has no unique solution. Also show that if $\neq$ fd 0 , then $B X=V$ has no solution.
Prove that the homogeneous system of equations ax + by + cz = 0 bx + cy + az = 0 cx + ay + bz = 0 has a nontrivial solution if and only if a³ + b³ + c³ - 3abc = 0
(a) Prove that solutions need not be unique for nonlinear initial-value problems by finding two solutions to $$ y \frac{d y}{d x}=x, \quad y(0)=0 $$ (b) Prove that solutions need not exist for nonlinear initial-value problems by showing that there is no solution for $$ y \frac{d y}{d x}=-x, \quad y(0)=0 $$
MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS
First-Order Differential Equations and Applications
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD