# Further Maths - Solving Systems of Equations Using Matricies > Source: https://ollybritton.com/notes/a-level/further-maths/topics/solving-systems-of-equations-matricies/ · Updated: 2020-09-25 · Tags: further-maths, latex-block-alt, school ##### What is the crucial idea that allows you to solve systems of equations using matricies?? If $$ A\left( \begin{matrix} x \\\\ y \\ z\end{matrix} \right) = v $$ then $$ \left( \begin{matrix} x \\\\ y \\ z\end{matrix} \right) = A^{-1}v $$ ##### How could you rewrite $$2x+3y = 4 \\\\ 4x-y = 7$$ as the product of two matricies?? $$ \left(\begin{matrix}2 & 3 \\\\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}4 \\ 7\end{matrix}\right) $$ ##### How could you solve this $$\left(\begin{matrix}2 & 3 \\\\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}4 \\ 7\end{matrix}\right)$$?? Multiply both left-hand sides of the equation by the inverse of the matrix. ##### How could you rewrite $$2x-6y+4z = 32 \\\\ 3x + 2y -9z = -49 \\ -2x + 4y + z = -3$$ (the jist is fine, not exact)?? $$ \left( \begin{matrix} 2 & -6 & 4 \\\\ 3 & 2 & -9 \\ -2 & 4 & 1\end{matrix} \right) = \left( \begin{matrix} 32 \\ -49 \\ -3\end{matrix} \right) $$ ##### What does it mean for a system of equations to be consistent?? There is at least one set of values that satisfies all three equations simulataneously. ##### What does it mean for a system of equations to be inconsistent?? There are no sets of values that satisfy all three equations simulatanaeously. ##### What is the geometric visualisation of three planes meeting at a point?? PHOTO ##### What does it mean in terms of the number of solutions for three planes meeting at a point?? The system of equations is consistent and has only one solution. ##### If a system of equations is consistent and only has one solution, then the planes form?? A point. PHOTO ##### What is the geometric visualisation of three planes forming a sheaf?? PHOTO ##### What does it mean in terms of the number of solutions and consistency for three planes forming a sheaf?? The system of equations is consistent and has infintely many solutions along a line. ##### If a system of equations is consistent and has infinitely many solutions along a line, then the planes form?? A sheaf. PHOTO ##### What does it mean in terms of the number of solutions and consistency for three planes being parallel and non-identical?? The system of equations is inconsistent and has no solutions. ##### What is the geometric visualisation of two or more planes being parallel and non-identical?? PHOTO ##### If a system of equations is inconsistent and has no solutions, then the planes can form what two scenarios?? * A prism * Two or more parallel, non-identical planes ##### What does it mean in terms of the number of solutions and consistency for three planes forming a prism?? The system of equations is inconsistent and has no solutions. ##### What is the geometric visualisation of two or more planes forming a prism?? PHOTO ##### What does it mean in terms of the number and solutions and consistency for three identical planes?? The system of equations is consistent and has infinitely many solutions along plane. ##### What is the geometric visualisation of three planes being the same?? PHOTO ##### If a system of equations is consistent and has infinitely many solutions along a plane, then the planes form?? A plane. They're all the same. PHOTO ##### If a system of equations is consistent but has a zero determinant, what are the two possible plane configurations?? * A sheaf * A plane ##### Hi! I'm a system of equations. My determinant is zero, and when you try and solve me manually you get GARBAGE. What's my consistency and plane configurations?? * Inconsistent * A triangular prism OR Parralel planes ##### Hi! I'm a system of equations. My determinant is non-zero. What's my consistency and plane configuration?? * Consistent * Planes meet at a point ##### Hi! I'm a system of equations. My determinant is zero and when you try and solve me manually you get $0 = 0$. What's my consistency and plane configurations?? * Consistent * A sheaf OR A it's the same plane --- Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt