Chapter 4

Determinants

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 (34)

Examples

Example 1Evaluate a $2\times2$ determinantworked
Example 2Evaluate a $2\times2$ determinant in $x$ worked
Example 3Evaluate a $3\times3$ determinantworked
Example 4A determinant of sines and cosines equals 0worked
Example 5Find $x$ from two equal determinantsworked
Example 6Area of the triangle $(3,8)$ ⁠, $(-4,2)$ ⁠, $(5,1)$ worked
Example 7Line through $A(1,3)$ ⁠, $B(0,0)$ and $k$ for area 3worked
Example 8Minor of the element 6worked
Example 9Minors and cofactors of a $2\times2$ determinantworked
Example 10Minors and cofactors of $a_{11}$ ⁠, $a_{21}$ worked
Example 11Minors and cofactors; check $a_{11}A_{31}+a_{12}A_{32}+a_{13}A_{33}=0$ worked
Example 12Find $\operatorname{adj}A$ for a $2\times2$ matrixworked
Example 13Verify $A(\operatorname{adj}A)=|A|I$ ⁠; find $A^{-1}$ worked
Example 14Verify $(AB)^{-1}=B^{-1}A^{-1}$ worked
Example 15 $A^2-4A+I=O$ ⁠; use it to find $A^{-1}$ worked
Example 16Solve two equations by the matrix methodworked
Example 17Solve three equations by the matrix methodworked
Example 18Three numbers from three conditionsworked
Example 19Use a given product to solve a systemworked

Exercise 4.1

Exercise 4.1 · Q1Evaluate a $2\times2$ determinantworked
Exercise 4.1 · Q2Evaluate two determinantsworked
Exercise 4.1 · Q3Show $|2A|=4|A|$ worked
Exercise 4.1 · Q4Show $|3A|=27|A|$ worked
Exercise 4.1 · Q5Evaluate four $3\times3$ determinantsworked
Exercise 4.1 · Q6Find $|A|$ worked
Exercise 4.1 · Q7Find $x$ from equal determinantsworked
Exercise 4.1 · Q8 (MCQ)Find $x$ from equal determinantsworked

Exercise 4.2

Exercise 4.2 · Q1Areas of three trianglesworked
Exercise 4.2 · Q2Show $A(a,b+c)$ ⁠, $B(b,c+a)$ ⁠, $C(c,a+b)$ are collinearworked
Exercise 4.2 · Q3Find $k$ for a triangle of area 4worked
Exercise 4.2 · Q4Equations of lines using determinantsworked
Exercise 4.2 · Q5 (MCQ)Find $k$ for a triangle of area 35worked

Exercise 4.3

Exercise 4.3 · Q1Minors and cofactors of $2\times2$ determinantsworked
Exercise 4.3 · Q2Minors and cofactors of $3\times3$ determinantsworked
Exercise 4.3 · Q3Evaluate using cofactors of the second rowworked
Exercise 4.3 · Q4Evaluate using cofactors of the third columnworked
Exercise 4.3 · Q5 (MCQ)Which expression equals $\Delta$ ⁠?worked

Exercise 4.4

Exercise 4.4 · Q1Adjoint of a $2\times2$ matrixworked
Exercise 4.4 · Q2Adjoint of a $3\times3$ matrixworked
Exercise 4.4 · Q3Verify $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I$ ⁠, $2\times2$ worked
Exercise 4.4 · Q4Verify $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I$ ⁠, $3\times3$ worked
Exercise 4.4 · Q5Inverse of a $2\times2$ matrixworked
Exercise 4.4 · Q6Inverse of a $2\times2$ matrixworked
Exercise 4.4 · Q7Inverse of an upper triangular matrixworked
Exercise 4.4 · Q8Inverse of a lower triangular matrixworked
Exercise 4.4 · Q9Inverse of a $3\times3$ matrixworked
Exercise 4.4 · Q10Inverse of a $3\times3$ matrix by the adjointworked
Exercise 4.4 · Q11Inverse of a matrix with $\cos\alpha$ and $\sin\alpha$ worked
Exercise 4.4 · Q12Verify $(AB)^{-1}=B^{-1}A^{-1}$ worked
Exercise 4.4 · Q13Show $A^2-5A+7I=O$ ⁠; hence find $A^{-1}$ worked
Exercise 4.4 · Q14Find $a$ and $b$ with $A^2+aA+bI=O$ worked
Exercise 4.4 · Q15Show $A^3-6A^2+5A+11I=O$ ⁠; hence find $A^{-1}$ worked
Exercise 4.4 · Q16Verify $A^3-6A^2+9A-4I=O$ ⁠; hence find $A^{-1}$ worked
Exercise 4.4 · Q17 (MCQ) $|\operatorname{adj}A|$ for a non-singular $3\times3$ matrixworked
Exercise 4.4 · Q18 (MCQ) $\det(A^{-1})$ for an invertible matrix of order 2worked

Exercise 4.5

Exercise 4.5 · Q1Is $x+2y=2$ ⁠, $2x+3y=3$ consistent?worked
Exercise 4.5 · Q2Is $2x-y=5$ ⁠, $x+y=4$ consistent?worked
Exercise 4.5 · Q3Is $x+3y=5$ ⁠, $2x+6y=8$ consistent?worked
Exercise 4.5 · Q4Consistency of a $3\times3$ system containing a constant $a$ worked
Exercise 4.5 · Q5Is $3x-y-2z=2$ ⁠, $2y-z=-1$ ⁠, $3x-5y=3$ consistent?worked
Exercise 4.5 · Q6Is $5x-y+4z=5$ ⁠, $2x+3y+5z=2$ ⁠, $5x-2y+6z=-1$ consistent?worked
Exercise 4.5 · Q7Solve $5x+2y=4$ ⁠, $7x+3y=5$ by the matrix methodworked
Exercise 4.5 · Q8Solve $2x-y=-2$ ⁠, $3x+4y=3$ by the matrix methodworked
Exercise 4.5 · Q9Solve $4x-3y=3$ ⁠, $3x-5y=7$ by the matrix methodworked
Exercise 4.5 · Q10Solve $5x+2y=3$ ⁠, $3x+2y=5$ by the matrix methodworked
Exercise 4.5 · Q11Solve $2x+y+z=1$ ⁠, $x-2y-z=\frac{3}{2}$ ⁠, $3y-5z=9$ by the matrix methodworked
Exercise 4.5 · Q12Solve $x-y+z=4$ ⁠, $2x+y-3z=0$ ⁠, $x+y+z=2$ by the matrix methodworked
Exercise 4.5 · Q13Solve $2x+3y+3z=5$ ⁠, $x-2y+z=-4$ ⁠, $3x-y-2z=3$ by the matrix methodworked
Exercise 4.5 · Q14Solve $x-y+2z=7$ ⁠, $3x+4y-5z=-5$ ⁠, $2x-y+3z=12$ by the matrix methodworked
Exercise 4.5 · Q15Find $A^{-1}$ and use it to solve a systemworked
Exercise 4.5 · Q16Onion, wheat and rice prices by the matrix methodworked

Miscellaneous Exercise

Miscellaneous · Q1A determinant independent of $\theta$ worked
Miscellaneous · Q2Evaluate a determinant in $\alpha$ ⁠, $\beta$ worked
Miscellaneous · Q3Find $(AB)^{-1}$ from $A^{-1}$ and $B$ worked
Miscellaneous · Q4Verify $[\operatorname{adj}A]^{-1}=\operatorname{adj}(A^{-1})$ and $(A^{-1})^{-1}=A$ worked
Miscellaneous · Q5Evaluate a determinant in $x$ ⁠, $y$ worked
Miscellaneous · Q6Evaluate another determinant in $x$ ⁠, $y$ worked
Miscellaneous · Q7Solve a system in $\frac1x$ ⁠, $\frac1y$ ⁠, $\frac1z$ worked
Miscellaneous · Q8 (MCQ)Inverse of a diagonal matrixworked
Miscellaneous · Q9 (MCQ)Where does $\det A$ lie?worked

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