@?further-maths

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 $$\left(\begin{matrix}2 & 3 \\ 4 & -1\end{matrix}\right)\left(\begin{matrix}x \ y\end{matrix}\right) = \left(\begin{matrix}4 \ 7\end{matrix}\right)$$?

What is the math definition of the inverse matrix of $M$?? The inverse matrix of $M$ is the matrix $M^{-1}$ such that $MM^{-1} = M^{-1}M = I$. What’s the wordy definition for an inverse matrix of $M$?? The matrix that when multiplied by the original matrix yields the identity matrix. What’s the formula for the inverse of matrix $$M = \left( \begin{matrix}a & b \\ c & d\end{matrix} \right)$$?? $$ \frac{1}{\text{ad - bc} M} \left( \begin{matrix}d & -b \\ -c & a\end{matrix} \right) $$

What is the determinant of a matrix?? A special value associated with a matrix. What does $|M|$ mean in the context of matricies?? The determinant of the matrix $M$. What does $\text{det} M$ mean?? The determinant of matrix $M$. What’s the formula for the determinant of a $2x2$ matrix?? $$ ad - bc $$ What is $M$ if $\text{det} M$ is $0$?? $M$ is a singular matrix. What is $M$ if $\text{det} M$ is not $0$?

See also: [[Further Maths - Determinants]]S [[Further Maths - Inverting Matricies]]S [[Further Maths - Solving Systems of Equations Using Matricies]]S [[Further Maths - Linear Transformations]]S If a matrix is $m \times n$, how many columns does it have?? $m$ columns. If a matrix is $m \times n$, how many rows does it have?? $n$ rows. What are the dimensions of a matrix?? $$ \text{rows} \times \text{columns} $$

The sole purpose of this entry being seperate is so that I don’t have to re-load the other [[Further Maths - Series]]S into Anki because it’s got some formatting issues that are down to underscores inside Latex formulas being recognised as italics rather than subscripts. How can you rewrite $\sum^{n}_{r=1} (r+4)^3$?? $$ \sum^{n+4}{r=1} r^3 + \sum{4}{r=1} r^3 $$ Why can you rewrite $\sum^{n}_{r=1} (r+4)^3$?? Because it’s the same as a $r^3$ sequence starting at $5$.

What is the formula for the sum of the cubes of the first $n$ natural numbers?? $$ \sum^{n}_{r=1} r^3 = \frac{1}{4}n^2(n+1)^2 $$ How could you rewrite $\sum^{n}_{r=1} r^3$?? $$ \frac{1}{4}n^2(n+1)^2 $$ What’s another way of expressing $\frac{1}{4}n^2(n+1)^2$?? $$ \sum^{n}_{r=1} r^3 $$ How could you rewrite $\sum^{n}_{r=1} 4r^2$?? $$ n^2(n+1)^2 $$ What’s an easy way for remembering the sum of cubes formula?? It’s the sum of the natural numbers formula squared. Backlinks [[Further Maths - Series]]S [[Further Maths - Syllabus]]S Metadata date: 2020-09-16 17:57 tags: - '@?

What is the formula for the sum of the squares of the $n$ natural numbers?? $$ \sum^{n}_{r=1} r^2 = \frac{1}{6}n(n+1)(2n+1) $$ How could you rewrite $\sum^{n}_{r=1} r^2$?? $$ \frac{1}{6}n(n+1)(2n+1) $$ What’s another way of expressing $\frac{1}{6}n(n+1)(2n+1)$?? $$ \sum^{n}_{r=1} r^2 $$ How could you rewrite $\sum^{n}_{r=1} 3r^2$?? $$ \frac{1}{3}n(n+1)(2n+1) $$ Backlinks [[Further Maths - Series]]S [[Further Maths - Syllabus]]S Metadata date: 2020-09-16 17:57 tags: - '@?further-maths' - '@?series' - '@?

Sums of Natural Numbers What are the natural numbers?? $1, 2, 3, 4, 5, 6\ldots$ What are the whole numbers?? $0, 1, 2, 3, 4, 5, 6\ldots$ What’s another name for the sum of the first $n$ natural numbers?? The triangle numbers. How can you visualise the sum of the first $n$ natural numbers?? As a triangle. How can you visualise the sum of the first $n$ natural numbers, doubled?? As a $n \times (n+1)$ square.

Series Core 1 [[Further Maths - Sums of Natural Numbers]]S [[Further Maths - Sums of Squares]]S [[Further Maths - Sums of Cubes]]S [[Further Maths - Series Tips and Tricks]]S [[Further Maths - Induction for Series]]S Core 2 [[Further Maths - The Method of Differences]]S Flashcards What is a series?? A sum of sequential terms. What is the notation for series?

Argand Diagrams Argand diagrams are a way of representing complex numbers by imagining them as points on a plane. In an Argand diagram: The $x$-axis is the “real” axis The $y$-axis is the “imaginary axis The complex number $z = x + yi$ can be represented on the diagram by the point $P(x, y)$ where $x$ and $y$ are coordinates. In other words: The horizontal position represents the real part of $z$. The vertical position represents the imaginary part of $z$.

Conjugates A pair of complex numbers with a different sign but otherwise the same values are called conjugates: $$ 3 + 4i 3 - 4i $$ What are complex conjugates?? A pair of complex numbers with real and imaginary parts equal in magnitude but opposite in sign: $a+bi$ and $a-bi$ What is a pair of complex numbers $a+bi$ and $a-bi$ called?? A complex conjugate. What is the result of multipling complex conjugates?? You get a real number.

Further Maths Syllabus Link to specification: https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-further-mathematics-specification.pdf Papers Paper 1 & 2: Core Pure Mathematics Core Pure 1 [[Further Maths - Complex Numbers]]S [[Further Maths - Cubics]]S [[Further Maths - Argand Diagrams]]S [[Further Maths - Conjugates]]S [[Further Maths - Series]]S [[Further Maths - Sums of Natural Numbers]]S [[Further Maths - Series Tips and Tricks]]S [[Further Maths - Sums of Squares]]S [[Further Maths - Sums of Cubes]]S [[Further Maths - Matricies]]S [[Further Maths - Determinants]]S [[Further Maths - Inverting Matricies]]S [[Further Maths - Solving Systems of Equations Using Matricies]]S [[Further Maths - Solving Systems of Equations Using Triangle Method]]S [[Further Maths - Volumes of Revolutions]]S [[Further Maths - Radians]]S [[Further Maths - Roots of Polynomials]]S [[Further Maths - Polar Form]]S [[Further Maths - Trigonometry Values]]S [[Further Maths - Odd and Even Functions]]S [[Further Maths - Loci in the Argand Diagram]]S [[Further Maths - Regions in the Argand Diagram]]S [[Further Maths - Linear Transformations]]S [[Further Maths - Induction]]S [[Further Maths - Induction for Series]]S [[Further Maths - Induction for Divisibility]]S [[Further Maths - Induction for Matricies]]S [[Further Maths - Vectors]]S [[Further Maths - Vector Equation of a Line]]S [[Further Maths - Vector Equation of a Plane]]S [[Further Maths - Dot Product]]S Core Pure 2 [[Further Maths - The Method of Differences]]S [[Further Maths - Partial Fractions]]S [[Further Maths - Maclaurin Series]]S [[Further Maths - Exponential Form of Complex Numbers]]S [[Further Maths - Trig Equations with Complex Numbers]]S [[Further Maths - Roots of Complex Numbers]]S [[Further Maths - Hyperbolic Functions]]S [[Further Maths - Mean Value of a Function]]S [[Further Maths - Improper Integrals]]S [[Further Maths - Integrating and Differentiating Inverse Trig Functions]]S [[Further Maths - Differential Equations]]S [[Further Maths - Coupled Differential Equations]]S Papers 3 & 4: Choices Further Pure 1 [[Further Maths - Inequalities]]S [[Further Maths - Conic Sections]]S [[Further Maths - T-formulae]]S [[Further Maths - Leibnitz's Theorem]]S [[Further Maths - L'Hôpital's Rule]]S [[Further Maths - Weierstrass Substitution]]S [[Further Maths - Taylor Series]]S [[Further Maths - Limits]]S [[Further Maths - Numerical Methods]]S [[Further Maths - Reducible Differential Equations]]S [[Further Maths - Cross Product]]S Decision Mathematics [[Further Maths - Bubblesort]]S [[Further Maths - Shuttlesort]]S [[Further Maths - Bin-packing]]S [[Further Maths - Graphs]]S [[Further Maths - Eulerian Trails]]S [[Further Maths - Hamiltonian Cycles]]S [[Further Maths - Kruskal's Algorithm]]S [[Further Maths - Prim's Algorithm]]S [[Further Maths - Polar Coordinates]]S [[Further Maths - Dijkstra's Algorithm]]S [[Further Maths - Route Inspection]]S [[Further Maths - Travelling Salesperson Problem]]S Info This page was last updated on 2022-02-21.