Flashcards · 10 cards

Matrices & linear transformations

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Rule
How do you multiply two matrices?
Answer
Across the row of the first, down the column of the second, multiplying in pairs and adding. $AB$ and $BA$ are usually different.
Row times columnMultiply across the row and down the column, then add
From the lesson: Matrix algebra and inverses
Rule
The inverse of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ ?
Answer
$\dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ . No inverse when $ad - bc = 0$ .
The inverseA⁻¹ = (1 ÷ (ad − bc)) × (d, −b / −c, a)
From the lesson: Matrix algebra and inverses
Know it
How do you solve two simultaneous equations with a matrix?
Answer
Write them as $A\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix}$ . Then $\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1}\begin{pmatrix} p \\ q \end{pmatrix}$ .
From the lesson: Matrix algebra and inverses
Rule
A $3 \times 3$ determinant along the top row?
Answer
$a(ei - fh) - b(di - fg) + c(dh - eg)$ : signs $+\;-\;+$ , each entry times the $2 \times 2$ left when you cover its row and column.
Along the top rowa(ei − fh) − b(di − fg) + c(dh − eg)
From the lesson: 3 × 3 determinants and Cramer's rule
Rule
Cramer's rule?
Answer
$x = \dfrac{\Delta_x}{\Delta}$ , $y = \dfrac{\Delta_y}{\Delta}$ , $z = \dfrac{\Delta_z}{\Delta}$ , where $\Delta_x$ is $\Delta$ with its $x$ -column replaced by the right-hand sides.
Cramer's rulex = Δx ÷ Δ, y = Δy ÷ Δ, z = Δz ÷ Δ
From the lesson: 3 × 3 determinants and Cramer's rule
Rule
What do the columns of a transformation matrix tell you?
Answer
Column 1 is the image of $(1, 0)$ ; column 2 is the image of $(0, 1)$ .
What the columns mean(1, 0) → (a, c) and (0, 1) → (b, d)
From the lesson: Linear transformations
Rule
The matrix of a rotation through $\theta$ anticlockwise about $O$ ?
Answer
$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ . Test it on $(1, 0)$ : the image must be $(\cos\theta, \sin\theta)$ .
Rotation through θi′ = (cos θ, sin θ) is column 1, j′ = (−sin θ, cos θ) is column 2
From the lesson: Linear transformations
Rule
The single matrix for " $N$ followed by $M$ "?
Answer
$MN$ : the transformation applied first goes on the right.
N followed by MThe combined matrix is MN
From the lesson: Linear transformations
Which method?
WAEC 2019 · Paper 2 · Q11
Given that $M : (x, y) \to (7y, 3x - y)$ and $N : (x, y) \to (2x - y, 5x + 3y)$ ,
write down the matrices $M$ and $N$ of the linear transformations;
find the image of $P(2, -3)$ under the linear transformation $N$ followed by $M$ ;
find the coordinates of the point $Q$ whose image is $Q'(2, 4)$ under the linear transformation $N$ .
How do you write $M$ and $N$ , and which product gives " $N$ followed by $M$ "?
Answer
Read the coefficients of each rule row by row: $M = \begin{pmatrix} 0 & 7 \\ 3 & -1 \end{pmatrix}$ . Then use $MN$ , with $N$ (applied first) on the right.
From the lesson: Linear transformations Try the question
Which method?
WAEC 2020 · Paper 2 · Q11 (a)
Find the positive value of $y$ for which $\begin{vmatrix} y & 1 & 0 \\ 1 & y & 1 \\ 0 & 1 & y \end{vmatrix} = 0$ (4 d.p.).
How do you expand this determinant?
Answer
Along the top row: $y(y^2 - 1) - 1(y - 0) + 0 = y^3 - 2y$ . Set that to 0 and solve.
From the lesson: 3 × 3 determinants and Cramer's rule Try the question