Flashcards · 10 cards

Matrices & determinants

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Rule
What is the order of a matrix?
Answer
Rows × columns, rows first. A matrix with 2 rows and 3 columns is $2 \times 3$ .
OrderRows first, then columns: this is 2 × 3
From the lesson: Adding and multiplying matrices
Know it
How do you add two matrices, or multiply a matrix by a number?
Answer
Adding: entry by entry, for matrices of the same order. A number multiplies every entry.
From the lesson: Adding and multiplying matrices
Rule
How do you multiply two matrices?
Answer
A row of the first times a column of the second: multiply in pairs and add. $AB$ and $BA$ are usually different.
Row times columnRow 1 of the first times column 1 of the second gives the top-left entry
From the lesson: Adding and multiplying matrices
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When can you work out $AB$ ?
Answer
When the number of columns of $A$ equals the number of rows of $B$ . A $2 \times 3$ times a $3 \times 1$ gives a $2 \times 1$ .
From the lesson: Adding and multiplying matrices
Rule
The determinant of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ ?
Answer
$ad - bc$ : the leading diagonal minus the other diagonal.
DeterminantLeading diagonal minus the other diagonal: ad − bc
From the lesson: Determinants and inverses
Rule
The inverse of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ ?
Answer
$\dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ : swap $a$ and $d$ , change the signs of $b$ and $c$ , divide by the determinant.
InverseSwap a and d, change the signs of b and c, divide by ad − bc
From the lesson: Determinants and inverses
Know it
When does a matrix have no inverse?
Answer
When its determinant is 0: the matrix is singular.
From the lesson: Determinants and inverses
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How do you solve $\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 10 \end{pmatrix}$ with matrices?
Answer
Multiply both sides on the left by the inverse of the matrix: $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}^{-1}\begin{pmatrix} 5 \\ 10 \end{pmatrix}$ .
From the lesson: Determinants and inverses
Rule
How do you find a $3 \times 3$ determinant?
Answer
Along the top row with the signs $+\;-\;+$ : each entry times the $2 \times 2$ determinant left when you cover its row and column.
3 × 3 determinantTop row, signs + − +, each times the 2 × 2 left over
From the lesson: Determinants and inverses
Which method?
WAEC 2019 · Paper 2 · Q13 (b)
Given the matrices $\mathbf A = \begin{pmatrix} 2 & 5 \\ -1 & -3 \end{pmatrix}$ and $\mathbf B = \begin{pmatrix} 3 & -2 \\ 4 & 1 \end{pmatrix}$ , find: (i) $\mathbf{BA}$ ; (ii) the determinant of $\mathbf{BA}$ .
In $\mathbf{BA}$ , whose rows do you use?
Answer
The first matrix's: rows of $\mathbf B$ times columns of $\mathbf A$ . Order matters, so $\mathbf{BA}$ is not $\mathbf{AB}$ .
From the lesson: Adding and multiplying matrices Try the question