Matrices

The Cayley–Hamilton theorem

Every quaternion $q = a + v$, with scalar part $a$ and vector part $v$, satisfies a quadratic equation with real coefficients. Since $v^2 = -\lvert v \rvert^2$ (see vectors), $$q^2 = a^2 + 2av + v^2 = a^2 - \lvert v \rvert^2 + 2av,$$and so $$q^2 - 2a\,q + \lvert q \rvert^2 = 0, \qquad \lvert q \rvert^2 = a^2 + \lvert v \rvert^2.$$A quaternion is a root of a real polynomial of degree 2, just as a complex number $z$ is a root of $z^2 - 2\operatorname{Re}(z)\,z + \lvert z \rvert^2 = 0$. In the Lectures on Quaternions of 1853 Hamilton went further and showed that certain linear functions of quaternions also satisfy a polynomial equation of their own, which he used to compute their inverses.

October 27, 2026 Read

ITA 2026 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2026 has 10 written questions. ITA does not publish official solutions for the written part (the 2nd phase answer key only covers the multiple-choice Portuguese questions); the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 01 Lattice points in the octahedron 231 02 Powers of a triangular matrix $n = 15$ 03 Parallelogram and inscribed circle 30 04 Red area of the target $\pi(\alpha^2 + 32\alpha + 528)$ 05 Four spheres in a larger sphere $\frac{22 + 9\sqrt{6}}{16}$ 06 Triangular matrices (inconsistent statement) see comment 07 Degree-4 polynomial and remainder $-1$, $\frac{1}{2}$, $1 \pm 2i$ 08 Increasing triples with sum 100 784 09 Tangents to the hyperbola through a point $y = \pm 2\sqrt{6}\,x - 8$; $(\pm\sqrt{6}, 4)$ 10 Intersecting circles and common area radii $2\sqrt{3}$ and $6$; area $10\pi - 12\sqrt{3}$ Question 01: Lattice points in the octahedron Statement Consider a regular octahedron centred at the origin, with three vertices at $(5, 0, 0)$, $(0, 5, 0)$ and $(0, 0, 5)$. How many points in the interior and on the surface of this solid have all integer coordinates?

October 28, 2026 Read

ITA 2026 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2026 had 48 multiple-choice questions: 12 in Mathematics, 12 in Physics, 12 in Chemistry and 12 in English. This post covers the mathematics questions, numbered 1 to 12 as in the exam. Exam conventions. $i^2 = -1$; $\bar{z}$ is the conjugate of $z$; $]a, b[$ is the open interval $a < x < b$.

October 28, 2026 Read

ITA 2025 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2025 has 10 written questions. ITA does not publish official solutions for the written part (the 2nd phase answer key only covers the multiple-choice Portuguese questions); the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 1 Remainder of the division by $x^2 - x + 1$ $a = -1$, $b = 3$ 2 Ellipse: maximum of $y/x$ $r = \sqrt{3}$ 3 Trigonometric system $\sin(\alpha + \beta) = \frac{5 + \sqrt{7}}{8}$ 4 Excircle $15\sqrt{3}$ 5 First digit of $3^{100}$ 5 6 Probability: fourth head on toss $n$ $n = 6$ and $n = 7$ 7 Polynomial with a double root and a right triangle $a = \pm\frac{3\sqrt[4]{2}}{2}$, $b = \mp\frac{1}{\sqrt[4]{2}}$ 8 Sum of determinants 1013 9 Integer matrices with $A^{-1} = A^T$ 3840 10 Projection of the cube perpendicular to the diagonal $4\sqrt{3}$ Question 1: The remainder of the division by x² − x + 1 Statement Find the real values $a$ and $b$ such that the polynomial $p(x) = x^{57} + ax^{14} + bx^7 + 1$, when divided by $x^2 - x + 1$, leaves remainder $2x + 1$.

October 28, 2026 Read

ITA 2022 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2022 had 70 multiple-choice questions: 15 in Physics, 15 in Portuguese, 10 in English, 15 in Mathematics and 15 in Chemistry. This post covers the mathematics questions, numbered 41 to 55 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i$ is the imaginary unit ($i^2 = -1$); $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment with endpoints $A$ and $B$, and $m(\overline{AB})$ is its length; $A\hat{O}B$ is the angle with vertex $O$ formed by the rays $\overrightarrow{OA}$ and $\overrightarrow{OB}$.

October 28, 2026 Read

Galois and Cayley: an algebra for things that are not numbers

In the small hours of 30 May 1832, in Paris, a young man of twenty spent the night writing. Évariste Galois had a duel arranged for dawn and knew he would probably not survive it. He wrote a long letter to his friend Auguste Chevalier summarising the ideas the Academy of Sciences had rejected or lost, and asked that it be shown to Jacobi and Gauss, “not on the truth, but on the importance of these theorems”. In the margin of a manuscript he left a sentence that became famous:

October 10, 2026 Read

FUVEST 2024 – 1st phase: Mathematics solved

Source: FUVEST – test of the 1st phase, version V and official template (rectified in 24/11/2023), available in the FUVEST collection 2024. The 1st phase of FUVEST 2024 (General Knowledge Test, 19/11/2023) had 90 questions from all disciplines. This post only contains 12 math questions. The numbering used here is that of Proof V; In versions K, Q, X and Z the same questions appear with other numbers. Template Question Subject Template 13 Matrices: product and social network B 31 Volume of cylinder and circular segment B 33 Percentage: cost variation A 34 Statistics: mode, median, mean and range A 35 Count: digital clock reflected B 48 Cobblestone Packaging D or E (both accepted) 49 Parallelogram and trigonometry C 59 Exponential and logarithmic function (inverse) D 60 Linear system and line graph E 63 Pythagorean Tree: geometric progression and logarithm C 84 Graph Reading: Homicide Rates E 85 Triangular and oblong numbers D Question 13: Matrices and connections in a social network Statement Matrices can be used to obtain information about a social network. To understand how this can be done, let’s consider as an example a small social network made up of 4 people: $P_1, P_2, P_3, P_4$. The matrix associated with this social network is the matrix $4 \times 4$:

September 27, 2026 Read