# Further Maths - Solving Systems of Equations Using Matrices > Source: https://ollybritton.com/notes/a-level/further-maths/topics/solving-systems-of-equations-matrices/ · Updated: 2020-09-25 · Tags: further-maths, latex-block-alt, school ### Setting up the matrix equation ##### What is the crucial idea that allows you to solve systems of equations using matrices?? If $$ A\left( \begin{matrix} x \\\\ y \\ z\end{matrix} \right) = v $$ then $$ \left( \begin{matrix} x \\\\ y \\ z\end{matrix} \right) = A^{-1}v $$ ^matrix-equation-solving-idea ##### How could you rewrite $$2x+3y = 4 \\\\ 4x-y = 7$$ as the product of two matrices?? $$ \left(\begin{matrix}2 & 3 \\\\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}4 \\ 7\end{matrix}\right) $$ ^rewrite-two-equations-as-matrices ##### 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. ^solve-by-premultiplying-inverse ##### How could you rewrite $$2x-6y+4z = 32 \\\\ 3x + 2y -9z = -49 \\ -2x + 4y + z = -3$$ (the gist is fine, not exact)?? $$ \left( \begin{matrix} 2 & -6 & 4 \\\\ 3 & 2 & -9 \\ -2 & 4 & 1\end{matrix} \right) = \left( \begin{matrix} 32 \\ -49 \\ -3\end{matrix} \right) $$ ^rewrite-three-equations-as-matrices ### Consistency ##### 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 simultaneously. ^consistent-system-definition ##### What does it mean for a system of equations to be inconsistent?? There are no sets of values that satisfy all three equations simultaneously. ^inconsistent-system-definition ### Geometric configurations of three planes ##### What is the geometric visualisation of three planes meeting at a point?? PHOTO ^three-planes-point-visualisation ##### 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. ^three-planes-point-solutions ##### If a system of equations is consistent and only has one solution, then the planes form?? A point. PHOTO ^consistent-unique-solution-forms-point ##### What is the geometric visualisation of three planes forming a sheaf?? PHOTO ^three-planes-sheaf-visualisation ##### 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 infinitely many solutions along a line. ^three-planes-sheaf-solutions ##### If a system of equations is consistent and has infinitely many solutions along a line, then the planes form?? A sheaf. PHOTO ^consistent-line-solutions-forms-sheaf ##### 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. ^parallel-planes-solutions ##### What is the geometric visualisation of two or more planes being parallel and non-identical?? PHOTO ^parallel-planes-visualisation ##### 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 ^inconsistent-no-solutions-configurations ##### 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. ^prism-planes-solutions ##### What is the geometric visualisation of two or more planes forming a prism?? PHOTO ^prism-planes-visualisation ##### 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. ^identical-planes-solutions ##### What is the geometric visualisation of three planes being the same?? PHOTO ^identical-planes-visualisation ##### 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 ^consistent-plane-solutions-forms-plane ### Using the determinant ##### If a system of equations is consistent but has a zero determinant, what are the two possible plane configurations?? * A sheaf * A plane ^zero-determinant-consistent-configurations ##### 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 Parallel planes ^zero-determinant-garbage-inconsistent ##### 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 ^non-zero-determinant-consistent ##### 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 ^zero-determinant-identity-consistent --- Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt