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Math  /  Algebra

QuestionSection 9.3 Follow-up Exercises Perform the following matrix operations wherever possible. 1−(4−258)−(31284)3−3(4−3−1−4)+8(1210−2−4)55(−210815)−3(20−25−1015)7(7−3)(48)9(3−2)(4−4−3)\begin{array}{l} 1-\left(\begin{array}{rr} 4 & -2 \\ 5 & 8 \end{array}\right)-\left(\begin{array}{rr} 3 & 12 \\ 8 & 4 \end{array}\right) \\ 3-3\left(\begin{array}{rr} 4 & -3 \\ -1 & -4 \end{array}\right)+8\left(\begin{array}{rr} 12 & 10 \\ -2 & -4 \end{array}\right) \\ 55\left(\begin{array}{rr} -2 & 10 \\ 8 & 15 \end{array}\right)-3\left(\begin{array}{rr} 20 & -25 \\ -10 & 15 \end{array}\right) \\ 7\left(\begin{array}{ll} 7 & -3 \end{array}\right)\binom{4}{8} \\ 9\left(\begin{array}{ll} 3 & -2 \end{array}\right)\left(\begin{array}{r} 4 \\ -4 \\ -3 \end{array}\right) \end{array} 2(5−8214)+(−6−210−4)−(−10521−8)43k(−ab−b2a)−2k(abb2a)6(7584)−8(4324)+6(1−82−4)8(1−2−3)(854)10(18−4−6)(23)\begin{array}{l} 2\left(\begin{array}{cc} 5 & -8 \\ 2 & 14 \end{array}\right)+\left(\begin{array}{rr} -6 & -2 \\ 10 & -4 \end{array}\right)-\left(\begin{array}{rr} -10 & 5 \\ 21 & -8 \end{array}\right) \\ 43 k\left(\begin{array}{cc} -a & b \\ -b & 2 a \end{array}\right)-2 k\left(\begin{array}{ll} a & b \\ b & 2 a \end{array}\right) \\ 6\left(\begin{array}{ll} 7 & 5 \\ 8 & 4 \end{array}\right)-8\left(\begin{array}{ll} 4 & 3 \\ 2 & 4 \end{array}\right)+6\left(\begin{array}{ll} 1 & -8 \\ 2 & -4 \end{array}\right) \\ 8\left(\begin{array}{ll} 1 & -2 \end{array}-3\right)\left(\begin{array}{l} 8 \\ 5 \\ 4 \end{array}\right) \\ 10\left(\begin{array}{lll} 18 & -4 & -6 \end{array}\right)\binom{2}{3} \end{array}

Studdy Solution

STEP 1

1. We are given a series of matrix operations to perform.
2. Matrix addition, subtraction, and scalar multiplication are the operations involved.
3. The matrices involved are of compatible dimensions for the operations specified.

STEP 2

1. Perform matrix subtraction for the first expression.
2. Perform scalar multiplication and matrix addition for the second expression.
3. Perform scalar multiplication and matrix subtraction for the third expression.
4. Perform matrix multiplication for the fourth expression.
5. Check the dimensions and perform matrix multiplication for the fifth expression.
6. Perform scalar multiplication and matrix addition/subtraction for the sixth expression.
7. Perform scalar multiplication and matrix subtraction for the seventh expression.
8. Perform scalar multiplication and matrix addition for the eighth expression.
9. Check the dimensions and perform matrix multiplication for the ninth expression.
10. Perform scalar multiplication and matrix multiplication for the tenth expression.

STEP 3

Perform matrix subtraction for the first expression:
1−(4−258)−(31284) 1 - \begin{pmatrix} 4 & -2 \\ 5 & 8 \end{pmatrix} - \begin{pmatrix} 3 & 12 \\ 8 & 4 \end{pmatrix}
First, subtract the matrices:
(4−258)−(31284)=(1−14−34) \begin{pmatrix} 4 & -2 \\ 5 & 8 \end{pmatrix} - \begin{pmatrix} 3 & 12 \\ 8 & 4 \end{pmatrix} = \begin{pmatrix} 1 & -14 \\ -3 & 4 \end{pmatrix}
Then, subtract this result from 1 (interpreted as subtracting each element from 1):
(1−11+141+31−4)=(0154−3) \begin{pmatrix} 1-1 & 1+14 \\ 1+3 & 1-4 \end{pmatrix} = \begin{pmatrix} 0 & 15 \\ 4 & -3 \end{pmatrix}

STEP 4

Perform scalar multiplication and matrix addition for the second expression:
3−3(4−3−1−4)+8(1210−2−4) 3 - 3 \begin{pmatrix} 4 & -3 \\ -1 & -4 \end{pmatrix} + 8 \begin{pmatrix} 12 & 10 \\ -2 & -4 \end{pmatrix}
First, perform scalar multiplication:
3(4−3−1−4)=(12−9−3−12) 3 \begin{pmatrix} 4 & -3 \\ -1 & -4 \end{pmatrix} = \begin{pmatrix} 12 & -9 \\ -3 & -12 \end{pmatrix}
8(1210−2−4)=(9680−16−32) 8 \begin{pmatrix} 12 & 10 \\ -2 & -4 \end{pmatrix} = \begin{pmatrix} 96 & 80 \\ -16 & -32 \end{pmatrix}
Then, perform matrix addition:
3−(12−9−3−12)+(9680−16−32) 3 - \begin{pmatrix} 12 & -9 \\ -3 & -12 \end{pmatrix} + \begin{pmatrix} 96 & 80 \\ -16 & -32 \end{pmatrix}
Interpreting 3 as a matrix with all elements 3:
(3333)−(12−9−3−12)+(9680−16−32)=(87922247) \begin{pmatrix} 3 & 3 \\ 3 & 3 \end{pmatrix} - \begin{pmatrix} 12 & -9 \\ -3 & -12 \end{pmatrix} + \begin{pmatrix} 96 & 80 \\ -16 & -32 \end{pmatrix} = \begin{pmatrix} 87 & 92 \\ 22 & 47 \end{pmatrix}

STEP 5

Perform scalar multiplication and matrix subtraction for the third expression:
55(−210815)−3(20−25−1015) 55 \begin{pmatrix} -2 & 10 \\ 8 & 15 \end{pmatrix} - 3 \begin{pmatrix} 20 & -25 \\ -10 & 15 \end{pmatrix}
First, perform scalar multiplication:
55(−210815)=(−110550440825) 55 \begin{pmatrix} -2 & 10 \\ 8 & 15 \end{pmatrix} = \begin{pmatrix} -110 & 550 \\ 440 & 825 \end{pmatrix}
3(20−25−1015)=(60−75−3045) 3 \begin{pmatrix} 20 & -25 \\ -10 & 15 \end{pmatrix} = \begin{pmatrix} 60 & -75 \\ -30 & 45 \end{pmatrix}
Then, perform matrix subtraction:
(−110550440825)−(60−75−3045)=(−170625470780) \begin{pmatrix} -110 & 550 \\ 440 & 825 \end{pmatrix} - \begin{pmatrix} 60 & -75 \\ -30 & 45 \end{pmatrix} = \begin{pmatrix} -170 & 625 \\ 470 & 780 \end{pmatrix}

STEP 6

Perform matrix multiplication for the fourth expression:
7(7−3)(48) 7 \begin{pmatrix} 7 & -3 \end{pmatrix} \binom{4}{8}
First, perform scalar multiplication:
7(7−3)=(49−21) 7 \begin{pmatrix} 7 & -3 \end{pmatrix} = \begin{pmatrix} 49 & -21 \end{pmatrix}
Then, perform matrix multiplication:
(49−21)(48)=49×4+(−21)×8=196−168=28 \begin{pmatrix} 49 & -21 \end{pmatrix} \binom{4}{8} = 49 \times 4 + (-21) \times 8 = 196 - 168 = 28

STEP 7

Check the dimensions and perform matrix multiplication for the fifth expression:
9(3−2)(4−4−3) 9 \begin{pmatrix} 3 & -2 \end{pmatrix} \begin{pmatrix} 4 \\ -4 \\ -3 \end{pmatrix}
The dimensions are not compatible for matrix multiplication (1x2 and 3x1), so this operation is not possible.

STEP 8

Perform scalar multiplication and matrix addition/subtraction for the sixth expression:
2(5−8214)+(−6−210−4)−(−10521−8) 2 \begin{pmatrix} 5 & -8 \\ 2 & 14 \end{pmatrix} + \begin{pmatrix} -6 & -2 \\ 10 & -4 \end{pmatrix} - \begin{pmatrix} -10 & 5 \\ 21 & -8 \end{pmatrix}
First, perform scalar multiplication:
2(5−8214)=(10−16428) 2 \begin{pmatrix} 5 & -8 \\ 2 & 14 \end{pmatrix} = \begin{pmatrix} 10 & -16 \\ 4 & 28 \end{pmatrix}
Then, perform matrix addition and subtraction:
(10−16428)+(−6−210−4)−(−10521−8) \begin{pmatrix} 10 & -16 \\ 4 & 28 \end{pmatrix} + \begin{pmatrix} -6 & -2 \\ 10 & -4 \end{pmatrix} - \begin{pmatrix} -10 & 5 \\ 21 & -8 \end{pmatrix}
=(10−6−16−24+1028−4)+(10−5−218) = \begin{pmatrix} 10 - 6 & -16 - 2 \\ 4 + 10 & 28 - 4 \end{pmatrix} + \begin{pmatrix} 10 & -5 \\ -21 & 8 \end{pmatrix}
=(4−181424)+(10−5−218) = \begin{pmatrix} 4 & -18 \\ 14 & 24 \end{pmatrix} + \begin{pmatrix} 10 & -5 \\ -21 & 8 \end{pmatrix}
=(14−23−732) = \begin{pmatrix} 14 & -23 \\ -7 & 32 \end{pmatrix}

STEP 9

Perform scalar multiplication and matrix subtraction for the seventh expression:
43k(−ab−b2a)−2k(abb2a) 43k \begin{pmatrix} -a & b \\ -b & 2a \end{pmatrix} - 2k \begin{pmatrix} a & b \\ b & 2a \end{pmatrix}
First, perform scalar multiplication:
43k(−ab−b2a)=(−43ka43kb−43kb86ka) 43k \begin{pmatrix} -a & b \\ -b & 2a \end{pmatrix} = \begin{pmatrix} -43ka & 43kb \\ -43kb & 86ka \end{pmatrix}
2k(abb2a)=(2ka2kb2kb4ka) 2k \begin{pmatrix} a & b \\ b & 2a \end{pmatrix} = \begin{pmatrix} 2ka & 2kb \\ 2kb & 4ka \end{pmatrix}
Then, perform matrix subtraction:
(−43ka43kb−43kb86ka)−(2ka2kb2kb4ka) \begin{pmatrix} -43ka & 43kb \\ -43kb & 86ka \end{pmatrix} - \begin{pmatrix} 2ka & 2kb \\ 2kb & 4ka \end{pmatrix}
=(−43ka−2ka43kb−2kb−43kb−2kb86ka−4ka) = \begin{pmatrix} -43ka - 2ka & 43kb - 2kb \\ -43kb - 2kb & 86ka - 4ka \end{pmatrix}
=(−45ka41kb−45kb82ka) = \begin{pmatrix} -45ka & 41kb \\ -45kb & 82ka \end{pmatrix}

STEP 10

Perform scalar multiplication and matrix addition for the eighth expression:
6(7584)−8(4324)+6(1−82−4) 6 \begin{pmatrix} 7 & 5 \\ 8 & 4 \end{pmatrix} - 8 \begin{pmatrix} 4 & 3 \\ 2 & 4 \end{pmatrix} + 6 \begin{pmatrix} 1 & -8 \\ 2 & -4 \end{pmatrix}
First, perform scalar multiplication:
6(7584)=(42304824) 6 \begin{pmatrix} 7 & 5 \\ 8 & 4 \end{pmatrix} = \begin{pmatrix} 42 & 30 \\ 48 & 24 \end{pmatrix}
8(4324)=(32241632) 8 \begin{pmatrix} 4 & 3 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} 32 & 24 \\ 16 & 32 \end{pmatrix}
6(1−82−4)=(6−4812−24) 6 \begin{pmatrix} 1 & -8 \\ 2 & -4 \end{pmatrix} = \begin{pmatrix} 6 & -48 \\ 12 & -24 \end{pmatrix}
Then, perform matrix addition and subtraction:
(42304824)−(32241632)+(6−4812−24) \begin{pmatrix} 42 & 30 \\ 48 & 24 \end{pmatrix} - \begin{pmatrix} 32 & 24 \\ 16 & 32 \end{pmatrix} + \begin{pmatrix} 6 & -48 \\ 12 & -24 \end{pmatrix}
=(42−3230−2448−1624−32)+(6−4812−24) = \begin{pmatrix} 42 - 32 & 30 - 24 \\ 48 - 16 & 24 - 32 \end{pmatrix} + \begin{pmatrix} 6 & -48 \\ 12 & -24 \end{pmatrix}
=(10632−8)+(6−4812−24) = \begin{pmatrix} 10 & 6 \\ 32 & -8 \end{pmatrix} + \begin{pmatrix} 6 & -48 \\ 12 & -24 \end{pmatrix}
=(16−4244−32) = \begin{pmatrix} 16 & -42 \\ 44 & -32 \end{pmatrix}

STEP 11

Check the dimensions and perform matrix multiplication for the ninth expression:
8(1−2)(854) 8 \begin{pmatrix} 1 & -2 \end{pmatrix} \begin{pmatrix} 8 \\ 5 \\ 4 \end{pmatrix}
The dimensions are not compatible for matrix multiplication (1x2 and 3x1), so this operation is not possible.

STEP 12

0 Perform scalar multiplication and matrix multiplication for the tenth expression:
10(18−4−6)(23) 10 \begin{pmatrix} 18 & -4 & -6 \end{pmatrix} \binom{2}{3}
First, perform scalar multiplication:
10(18−4−6)=(180−40−60) 10 \begin{pmatrix} 18 & -4 & -6 \end{pmatrix} = \begin{pmatrix} 180 & -40 & -60 \end{pmatrix}
Then, perform matrix multiplication:
(180−40−60)(23) \begin{pmatrix} 180 & -40 & -60 \end{pmatrix} \binom{2}{3}
The dimensions are not compatible for matrix multiplication (1x3 and 2x1), so this operation is not possible.

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