General ScienceMadrid 2026Question 2A
X-linked inheritance
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Problem statement
Given the real matrix $A=\begin{pmatrix}1&\lambda-2&\lambda\\0&1&\lambda-1\\1&0&1\end{pmatrix}$, find:
a) (1 mark) Discuss the rank of $A$ as a function of the parameter $\lambda$.
b) (1 mark) For the case $\lambda=2$, solve the matrix equation $A^2-AX=I$, where $I$ is the identity matrix of order $3$.
Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Mathematics · Question 1. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.What this lesson answers
- What are we asked, and what can we anticipate without calculating?
- Why does the determinant decide whether the rank is 3?
- If the determinant is zero, why is the rank 2 and not 1?
- Why can't we “divide by A”, and on which side do we multiply?
- Where does the inverse come from, and what matrix is X?
- Another way: row reduction for the rank, and the equation without simplifying
- The answers and how to check them
- Common mistakes to avoid
- Exam tips
- If you want top marks
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