Chapter 3

Matrices

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (24)

Examples

Example 1Factory workers as a matrixworked
Example 2Possible orders of a matrix with 8 elementsworked
Example 3Construct a $3\times2$ matrix from $a_{ij}=\frac12|i-3j|$ worked
Example 4Equal matrices: find the unknownsworked
Example 5Find $a, b, c, d$ from a matrix equationworked
Example 6Find $A+B$ worked
Example 7Find $2A-B$ worked
Example 8Find $X$ with $2A+3X=5B$ worked
Example 9Find $X$ and $Y$ from $X+Y$ and $X-Y$ worked
Example 10Find $x$ and $y$ from a matrix equationworked
Example 11Two farmers' rice salesworked
Example 12Find $AB$ worked
Example 13Find $AB$ and $BA$ ⁠; show $AB\ne BA$ worked
Example 14 $AB\ne BA$ for two $2\times2$ matricesworked
Example 15 $AB=O$ with $A\ne O$ ⁠, $B\ne O$ worked
Example 16Find $A(BC)$ and $(AB)C$ ⁠; show they are equalworked
Example 17Verify $(A+B)C=AC+BC$ worked
Example 18Show $A^3-23A-40I=O$ worked
Example 19Election campaign costsworked
Example 20Verify $(A')'=A$ ⁠, $(A+B)'=A'+B'$ ⁠, $(kB)'=kB'$ worked
Example 21Verify $(AB)'=B'A'$ worked
Example 22Symmetric plus skew symmetricworked
Example 23Powers of the rotation matrix, by inductionworked
Example 24For symmetric $A, B$ ⁠: $AB$ symmetric iff $AB=BA$ worked
Example 25Find $D$ with $CD-AB=O$ worked

Exercise 3.1

Exercise 3.1 · Q1Order, elements and entries of a matrixworked
Exercise 3.1 · Q2Possible orders with 24 or 13 elementsworked
Exercise 3.1 · Q3Possible orders with 18 or 5 elementsworked
Exercise 3.1 · Q4Construct $2\times2$ matrices from $a_{ij}$ rulesworked
Exercise 3.1 · Q5Construct $3\times4$ matrices from $a_{ij}$ rulesworked
Exercise 3.1 · Q6Find $x, y, z$ from equal matricesworked
Exercise 3.1 · Q7Find $a, b, c, d$ from equal matricesworked
Exercise 3.1 · Q8 (MCQ)When is $A=[a_{ij}]_{m\times n}$ a square matrix?worked
Exercise 3.1 · Q9 (MCQ)Values that make two matrices equalworked
Exercise 3.1 · Q10 (MCQ)How many $3\times3$ matrices with entries 0 or 1?worked

Exercise 3.2

Exercise 3.2 · Q1Find $A+B$ ⁠, $A-B$ ⁠, $3A-C$ ⁠, $AB$ ⁠, $BA$ worked
Exercise 3.2 · Q2Compute sums of matricesworked
Exercise 3.2 · Q3Compute products of matricesworked
Exercise 3.2 · Q4Find $A+B$ ⁠, $B-C$ ⁠; verify $A+(B-C)=(A+B)-C$ worked
Exercise 3.2 · Q5Compute $3A-5B$ worked
Exercise 3.2 · Q6Simplify a trigonometric matrix sumworked
Exercise 3.2 · Q7Find $X$ and $Y$ worked
Exercise 3.2 · Q8Find $X$ from $2X+Y$ worked
Exercise 3.2 · Q9Find $x$ and $y$ worked
Exercise 3.2 · Q10Solve for $x, y, z, t$ worked
Exercise 3.2 · Q11Find $x$ and $y$ from a column equationworked
Exercise 3.2 · Q12Find $x, y, z, w$ worked
Exercise 3.2 · Q13Show $F(x)F(y)=F(x+y)$ worked
Exercise 3.2 · Q14Show that two products differworked
Exercise 3.2 · Q15Find $A^2-5A+6I$ worked
Exercise 3.2 · Q16Prove $A^3-6A^2+7A+2I=O$ worked
Exercise 3.2 · Q17Find $k$ with $A^2=kA-2I$ worked
Exercise 3.2 · Q18Show $I+A=(I-A)$ times a rotation matrixworked
Exercise 3.2 · Q19Trust fund bondsworked
Exercise 3.2 · Q20Bookshop stock valueworked
Exercise 3.2 · Q21 (MCQ)When is $PY+WY$ defined?worked
Exercise 3.2 · Q22 (MCQ)Order of $7X-5Z$ worked

Exercise 3.3

Exercise 3.3 · Q1Transposesworked
Exercise 3.3 · Q2Verify $(A+B)'=A'+B'$ and $(A-B)'=A'-B'$ worked
Exercise 3.3 · Q3Verify $(A+B)'=A'+B'$ and $(A-B)'=A'-B'$ given $A'$ worked
Exercise 3.3 · Q4Find $(A+2B)'$ given $A'$ and $B$ worked
Exercise 3.3 · Q5Verify $(AB)'=B'A'$ worked
Exercise 3.3 · Q6Verify $A'A=I$ worked
Exercise 3.3 · Q7Symmetric and skew symmetric matricesworked
Exercise 3.3 · Q8 $A+A'$ symmetric, $A-A'$ skew symmetricworked
Exercise 3.3 · Q9Find $\frac12(A+A')$ and $\frac12(A-A')$ worked
Exercise 3.3 · Q10Symmetric plus skew symmetric, four matricesworked
Exercise 3.3 · Q11 (MCQ) $AB-BA$ for symmetric $A, B$ worked
Exercise 3.3 · Q12 (MCQ)Find $\alpha$ from $A+A'=I$ worked

Exercise 3.4

Exercise 3.4 · Q1 (MCQ)When are A and B inverses?worked

Miscellaneous Exercise

Miscellaneous · Q1 $AB-BA$ is skew symmetricworked
Miscellaneous · Q2 $B'AB$ symmetric or skew symmetricworked
Miscellaneous · Q3Find $x, y, z$ with $A'A=I$ worked
Miscellaneous · Q4Find $x$ from a row-matrix-column productworked
Miscellaneous · Q5Show $A^2-5A+7I=O$ worked
Miscellaneous · Q6Find $x$ from a product equal to $O$ worked
Miscellaneous · Q7Manufacturer's revenue, cost and profitworked
Miscellaneous · Q8Find $X$ from $XA=B$ worked
Miscellaneous · Q9 (MCQ)Condition from $A^2=I$ worked
Miscellaneous · Q10 (MCQ)Both symmetric and skew symmetricworked
Miscellaneous · Q11 (MCQ) $(I+A)^3-7A$ when $A^2=A$ worked

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