Matrices & linear transformations

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1. 1Matrix algebra and inversesAdding and multiplying matrices, matrix equations with the identity, the inverse of a 2 × 2 matrix, and solving two simultaneous equations with it.Done18 min
2. 23 × 3 determinants and Cramer's ruleEvaluating a 3 × 3 determinant along the top row, solving equations for an unknown in a determinant, and solving three simultaneous equations by Cramer's rule.Done18 min
3. 3Linear transformationsThe matrix of a linear transformation, rotation about the origin, images and pre-images, one transformation followed by another, and finding a matrix from the images of two points.Done22 min

Past questions (31)

From 2008–2023: 30 WAEC, 1 NECO. Theory questions first, then objective, newest first.

Theory (24)

• WAEC 2023 · Paper 2 · Q10Calculate $\begin{vmatrix} 3 & 5 & -4 \\ 6 & -3 & -5 \\ -2 & 2 & 1 \end{vmatrix}$.
• WAEC 2022 · Paper 2 · Q11Write down the matrix $M$ of the linear transformation defined by $T : (x, y) \to (3x + y, -2x + 3y)$.
• WAEC 2020 · Paper 2 · Q11Find the positive value of $y$ for which $\begin{vmatrix} y & 1 & 0 \\ 1 & y & 1 \\ 0 & 1 & y \end{vmatrix} = 0$ (4 d.p.).
• WAEC 2020 · Paper 2 · Q2Given that $P = \begin{pmatrix} 3 & 4 \\ 5 & 6 \end{pmatrix}$ and $Q = \begin{pmatrix} -2 & 5 \\ -3 & 1 \end{pmatrix}$, find $PQ^{-1}$, where …
• WAEC 2019 · Paper 2 · Q11Given that $M : (x, y) \to (7y, 3x - y)$ and $N : (x, y) \to (2x - y, 5x + 3y)$,
• WAEC 2019 · Paper 2 · Q11Using determinants, solve the following equations simultaneously: $5x - 6y + 4z = 15$ $7x + 4y - 3z = 19$ $2x + y + 6z = 46$
• WAEC 2019 · Paper 2 · Q11Two linear transformations $P$ and $Q$ in the $Oxy$ plane are defined by $P : (x, y) \to (x + 2y, x)$ and $Q : (x, y) \to (x + y, -x + 2y)$. …
• WAEC 2018 · Paper 2 · Q1If $\begin{vmatrix} x - 3 & -4 & 3 \\ 5 & 2 & 2 \\ 2 & -4 & 6 - x \end{vmatrix} = -24$, find the values of $x$.
• WAEC 2018 · Paper 2 · Q11Given the matrix $A = \begin{pmatrix} 3 & 8 \\ 5 & -2 \end{pmatrix}$, find its inverse.
• WAEC 2018 · Paper 2 · Q3Given that $B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $B^2 + 3B + 2I = 3N$, where $I$ is the $2 \times 2$ unit matrix, find the matrix …
• WAEC 2017 · Paper 2 · Q9Given that $F = \begin{pmatrix} 3 & 4 & 2 \\ 0 & -1 & 2 \\ 2 & 1 & -2 \end{pmatrix}$:
• WAEC 2016 · Paper 2 · Q9Without using mathematical tables or a calculator, evaluate $\dfrac{\frac32\log27 - 3\log5\sqrt5}{\log0.6}$.
• WAEC 2014 · Paper 2 · Q10Express $\dfrac{5 + \sqrt2}{3 - \sqrt2} - \dfrac{5 - \sqrt2}{3 + \sqrt2}$ in the form $a + b\sqrt2$.
• WAEC 2014 · Paper 2 · Q10If $P = \begin{pmatrix} 4 & 2 \\ c & 3 \end{pmatrix}$, $Q = \begin{pmatrix} 6 & 2 \\ 4 & d \end{pmatrix}$ and $PQ = QP$, find the values of $c$ …
• WAEC 2013 · Paper 2 · Q9Differentiate $\dfrac{x^2 + 1}{(x + 1)^2}$ with respect to $x$.
• WAEC 2012 · Paper 2 · Q12Write down the matrices $P$ and $Q$ of the transformations $P : (x, y) \to (3x - 4y, -x)$ and $Q : (x, y) \to (y, -2x + y)$.
• WAEC 2011 · Paper 2 · Q10If $\mathbf A = \begin{pmatrix} -2 & 5 \\ 4 & 3 \end{pmatrix}$ and $\mathbf B = \begin{pmatrix} 3 & 1 \\ 2 & 3 \end{pmatrix}$, find the values …
• WAEC 2011 · Paper 2 · Q11The images of points $(2, 3)$ and $(4, 5)$ under a linear transformation $\mathbf A$ are $(3, 4)$ and $(5, 6)$ respectively. Find the:
• WAEC 2011 · Paper 2 · Q11Solve for $x$, $y$ and $z$ in the equations: $3x + 5y - 4z = -5$; $6x + 3y - 5z = 26$; $-2x + 2y + z = -11$.
• WAEC 2009 · Paper 2 · Q11Evaluate $\displaystyle\int_1^2 \frac{x}{\sqrt{5 - x^2}}\,dx$.
• WAEC 2009 · Paper 2 · Q12Use the trapezium rule with five ordinates to evaluate $\displaystyle\int_0^1 \frac{x}{1 + x^2}\,dx$, correct to four significant figures.
• WAEC 2009 · Paper 2 · Q12Two linear transformations are defined by $P : (x, y) \to (-2x + 3y,\ 5x - 4y)$ and $T : (x, y) \to (-3x - 4y,\ 6x - 5y)$. Find the inverse of …
• WAEC 2008 · Paper 2 · Q12The point $P(3, -5)$ is rotated through an angle of $60^\circ$ anticlockwise about the origin. (i) Obtain the matrix for the rotation. (ii) Find …
• WAEC 2008 · Paper 2 · Q12Given that $M\begin{pmatrix} 2 & 9 \\ 8 & 6 \end{pmatrix} = \begin{pmatrix} 14 & 3 \end{pmatrix}$, find the matrix $M$.

Objective (7)

• NECO 2023 · Paper 1 · Q13If $\begin{pmatrix} 3 & -1 \\ 1 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 13 \\ 7 \end{pmatrix}$, find the value …
• WAEC 2023 · Paper 1 · Q21A linear transformation on the $oxy$ plane is defined by $P: (x, y) \rightarrow (2x + y, -2y)$. Find $P^2$.
• WAEC 2023 · Paper 1 · Q24Given that $P = \begin{pmatrix} x & 4 \\ 3 & 7 \end{pmatrix}$, $Q = \begin{pmatrix} x & 3 \\ 1 & 2x \end{pmatrix}$ and the determinant of $Q$ …
• WAEC 2022 · Paper 1 · Q11If $\begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix}\begin{pmatrix} -6 \\ k \end{pmatrix} = \begin{pmatrix} 3 \\ -26 \end{pmatrix}$, find the value …
• WAEC 2022 · Paper 1 · Q12A linear transformation T is defined by $T: (x, y) \to (3x - y, x + 4y)$. Find the image of $(2, -1)$ under T.
• WAEC 2020 · Paper 1 · Q2Find the inverse of $\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}$.
• WAEC 2020 · Paper 1 · Q5If $\begin{pmatrix} p + q & 1 \\ 0 & p - q \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 0 & 8 \end{pmatrix}$, find the values of $p$ and $q$.