Matrix Arithmetic, Inverse & Sim Equations, Transformations etc
What are matrices and how do we calculate with them?
.png)
On this page you can find a set of videos, created by Dr James Tanton, that provide an intuitive, conceptual insight into what matrices are, why they are useful and how we calculate with them. Calculating a matrix's inverse and solving simultaneous equations using matrices are also covered [note: the film 'The Matrix' (pause the clip at 34, 58, 66/67, 68/69 and 80 seconds to see 'the matrix'!) was not created by James, nor us! Tok: what's the link between the film and this topic?]
How/why, conceptually, do matrices define transformations. An in-depth look at the geometrical meaning of a matrix's determinant and an introduction to Adjacency matrices follow at the end.
Matrices 1: A Natural Appearance/utility
Adjacency Matrices (an introduction) - another Natural use for matrices
Matrices 2: Scalar Multiplication - Naturally!
Matrices 3: Matrix Addition - Naturally!
Matrices 4: Matrix Multiplication .. Naturally!
Matrices 5: Matrix Inverses . . Not as Naturally! (but necessary!)
Matrices 6: Formal Associativity of Matrices
Associative: re-arranging which items you bracket won't change the outcome, as with equations that only contain multiplication or only addition, but not in a calculation with a mix of multiplications and additions.
Mutiplication and addition do not, combined, show associativity e.g. (1+1)x2 ≠ 1+(1x2) [not associative] but (3x4)x5 = 3x(4x5) [associative].
Matrices 7: Solving Simultaneous Equations
Practice question
Simultaneous equation question
Q1) Solve the following (unfriendly) system of equations using matrix methods.
\(\begin{array}{l}3x + 1.7y = e\\3x - 3.3y = \pi \end{array} % MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIZa % GaamiEaiabgUcaRiaaigdacaGGUaGaaG4naiaadMhacqGH9aqpcaWG % LbaabaGaaG4maiaadIhacqGHsislcaaIZaGaaiOlaiaaiodacaWG5b % Gaeyypa0JaeqiWdahaaaa!4647! \)
SOLUTION
Set the matrix: \(A = \left( {\begin{array}{*{20}{c}}3&{1.7}\\3&{ - 3.3}\end{array}} \right) % MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 % da9maabmaabaqbaeqabiGaaaqaaiaaiodaaeaacaaIXaGaaiOlaiaa % iEdaaeaacaaIZaaabaGaeyOeI0IaaG4maiaac6cacaaIZaaaaaGaay % jkaiaawMcaaaaa!401A! \)
Then the system of equations reads:
\(A\left( {\begin{array}{*{20}{c}}x\\y\end{array}} \right) = \left( {\begin{array}{*{20}{c}}e\\\pi \end{array}} \right) % MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaabm % aabaqbaeqabiqaaaqaaiaadIhaaeaacaWG5baaaaGaayjkaiaawMca % aiabg2da9maabmaabaqbaeqabiqaaaqaaiaadwgaaeaacqaHapaCaa % aacaGLOaGaayzkaaaaaa!3F8E! \)
The matrix is invertible and so the system has the unique solution:
\(\left( {\begin{array}{*{20}{c}}x\\y\end{array}} \right) = {A^{ - 1}}\left( {\begin{array}{*{20}{c}}e\\\pi \end{array}} \right) = - \frac{1}{{15}}\left( {\begin{array}{*{20}{c}}{ - 3.3}&{ - 1.7}\\{ - 3}&3\end{array}} \right)\left( {\begin{array}{*{20}{c}}e\\\pi \end{array}} \right) % MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa % qabeGabaaabaGaamiEaaqaaiaadMhaaaaacaGLOaGaayzkaaGaeyyp % a0JaamyqamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmaabaqbae % qabiqaaaqaaiaadwgaaeaacqaHapaCaaaacaGLOaGaayzkaaGaeyyp % a0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGymaiaaiwdaaaWaaeWaae % aafaqabeGacaaabaGaeyOeI0IaaG4maiaac6cacaaIZaaabaGaeyOe % I0IaaGymaiaac6cacaaI3aaabaGaeyOeI0IaaG4maaqaaiaaiodaaa % aacaGLOaGaayzkaaWaaeWaaeaafaqabeGabaaabaGaamyzaaqaaiab % ec8aWbaaaiaawIcacaGLPaaaaaa!5416! \)
which gives:
\(\begin{array}{l}x = \frac{1}{{15}}\left( {3.3e + 1.7\pi } \right)\\y = \frac{1}{5}\left( {e - \pi } \right)\end{array} % MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4b % Gaeyypa0ZaaSaaaeaacaaIXaaabaGaaGymaiaaiwdaaaWaaeWaaeaa % caaIZaGaaiOlaiaaiodacaWGLbGaey4kaSIaaGymaiaac6cacaaI3a % GaeqiWdahacaGLOaGaayzkaaaabaGaamyEaiabg2da9maalaaabaGa % aGymaaqaaiaaiwdaaaWaaeWaaeaacaWGLbGaeyOeI0IaeqiWdahaca % GLOaGaayzkaaaaaaa!4C5A! \)
Bonus Material: Matrix Multiplication is Associative
Associative: re-arranging which items you bracket won't change the outcome, as with equations that only contain multiplication or only addition, but not in a calculation with a mix of multiplications and additions.
Mutiplication and addition do not, combined, show associativity e.g. (1+1)x2 ≠ 1+(1x2) [not associative] but (3x4)x5 = 3x(4x5) [associative].
IB exam style questions (subscription required for access)
Matrix Arithemetic, Determinant, Inverse and simultaneous equations :- use these Exam style questions to test if you feel ready, or what you need to review, ahead of moving on to the new topic or your end of unit test/mocks/final exams!
Matrices 7: Geometric Transformations
Optional Extra: A Full Understanding of one-dimensional Transformations
Click no the "eye" icon 
below to view this video.
Some standard Transformations: Reflections, Enlargements/Dilations, Rotations
Identifying Transformations with NO Memorization
Matrices 8: the Geometric Meaning of Determinants
Deeper: Oriented Areas and Negative Determinants
IB exam style questions (subscription required for access)
Matrix Transformations , use these Exam style questions to test if you feel ready, or what you need to review, ahead of moving on to the new topic or your end of unit test/mocks/final exams!
SOLUTIONS to exam style questions (including a guideline markscheme)