Matrix Algebra Tutor - Worksheet 9 â Matrix Determinants
Algebra 2 – Matrix Algebra Tutor Worksheet 9 – Matrix Determinants
1 © MathTutorDVD.com
Algebra 2 – Matrix Algebra Tutor - Worksheet 9 – Matrix Determinants 1. Find the determinant of this matrix. 𝐴=[
5 3 ] −8 2
2. Find the determinant of this matrix. 12 −3 𝐴=[ ] −7 18
3. Find the determinant of this matrix. 𝐴=[
33 −62 ] −11 37
2 © MathTutorDVD.com
4. Find the determinant of this matrix. 𝐴=[
−2 10
−13 ] −23
5. Find the determinant of this matrix. 𝐴=[
3 6 ] 2 7
6. Find the determinant of this matrix. −6 −6 𝐴 = [−4 −3 −5 −6
3 © MathTutorDVD.com
0 4] 6
7. Find the determinant of this matrix. −5 𝐴 = [−2 3
−1 4 2 −3] 4 6
8. Find the determinant of this matrix. 3 7 6 𝐴 = [ 4 9 5] 3 8 7
4 © MathTutorDVD.com
9. Find the determinant of this matrix. 1 −5 6 𝐴 = [4 −5 −4] 0 5 −6
10. Find the determinant of this matrix. 4 𝐴 = [−3 6
−1 2 1 −2] −1 −3
5 © MathTutorDVD.com
11. Find the determinant of this matrix. 6 −6 4 𝐴 = [3 2 −2] 3 5 −1
12. Find the determinant of this matrix. 3 4 2 𝐴 = [ 5 2 4] 2 3 6
6 © MathTutorDVD.com
13. Find the determinant of this matrix. 6 2 5 𝐴 = [ 3 2 2] 2 8 9
14. Find the determinant of this matrix. −5 4 −3 𝐴 = [−4 6 −4] 7 8 −2
7 © MathTutorDVD.com
Answers - Algebra 2 – Matrix Algebra Tutor - Worksheet 9 – Matrix Determinants Use the following formula find the determinant of a 2 by 2 matrix: 𝐴=[
𝑎 𝑐
𝑏 ] , 𝑑𝑒𝑡𝐴 = 𝑎 ∙ 𝑑 − 𝑏 ∙ 𝑐 𝑑
1. Find the determinant of this matrix. 𝐴=[
5 3 ] −8 2
𝑑𝑒𝑡𝐴 = 5 ∙ 2 − 3(−8) = 10 + 24 = 34 Answer: 34
2. Find the determinant of this matrix. 12 −3 𝐴=[ ] −7 18 𝑑𝑒𝑡𝐴 = 12 ∙ 18 − (−3)(−7) = 216 − 21 = 195 Answer: 195
3. Find the determinant of this matrix. 𝐴=[
33 −62 ] −11 37
𝑑𝑒𝑡𝐴 = 33 ∙ 37 − (−11)(−62) = 1221 − 682 = 539 Answer: 539
8 © MathTutorDVD.com
4. Find the determinant of this matrix. 𝐴=[
−2 10
−13 ] −23
𝑑𝑒𝑡𝐴 = (−2)(−23) − (−13)(10) = 46 + 130 = 176 Answer: 176
5. Find the determinant of this matrix. 𝐴=[
3 6 ] 2 7
𝑑𝑒𝑡𝐴 = 3 ∙ 7 − 6 ∙ 2 = 21 − 12 = 9 Answer: 9
Use the following formula to find the determinant of a 3 by 3 matrix: 𝑎 𝐴 = [𝑑 𝑔 𝑒 𝑑𝑒𝑡𝐴 = 𝑎 ∙ 𝑑𝑒𝑡 [ ℎ
𝑏 𝑒 ℎ
𝑐 𝑓] 𝑖
𝑑 𝑓 ] − 𝑏 ∙ 𝑑𝑒𝑡 [ 𝑔 𝑖
𝑓 𝑑 ] + 𝑐 ∙ 𝑑𝑒𝑡 [ 𝑔 𝑖
𝑒 ] ℎ
6. Find the determinant of this matrix. −6 −6 𝐴 = [−4 −3 −5 −6 𝑑𝑒𝑡𝐴 = −6 ∙ 𝑑𝑒𝑡 [
−3 4 −4 ] − (−6) ∙ 𝑑𝑒𝑡 [ −6 6 −5 9
© MathTutorDVD.com
0 4] 6 4 −4 −3 ] + 0 ∙ 𝑑𝑒𝑡 [ ] 6 −5 −6
= −6[(−3)(6) − (−6)(4)] + 6[(−4)(6) − (−5)(4)] + 0[(−4)(−6) − (−5)(−3)] −6[−18 + 24] + 6[−24 + 20] −6[6] + 6[−4] = −36 − 24 = −60 Answer: −60
7. Find the determinant of this matrix. −5 𝐴 = [−2 3 𝑑𝑒𝑡𝐴 = −5 ∙ 𝑑𝑒𝑡 [
−1 4 2 −3] 4 6
2 −3 −2 ] − (−1) ∙ 𝑑𝑒𝑡 [ 4 6 3
−3 −2 2 ] + 4 ∙ 𝑑𝑒𝑡 [ ] 6 3 4
= −5[(2)(6) − (−3)(4)] + [(−2)(6) − (−3)(3)] + 4[(−2)(4) − (2)(3)] = −5[12 + 12] + [−12 + 9] + 4[−8 − 6] = −5[24] + [−3] + 4[−14] = −120 − 3 − 56 = −179 Answer: −179
10 © MathTutorDVD.com
8. Find the determinant of this matrix. 3 7 6 𝐴 = [ 4 9 5] 3 8 7 𝑑𝑒𝑡𝐴 = 3 ∙ 𝑑𝑒𝑡 [
4 9 9 5 4 5 ] − 7 ∙ 𝑑𝑒𝑡 [ ] + 6 ∙ 𝑑𝑒𝑡 [ ] 3 8 8 7 3 7
= 3[(9)(7) − (5)(8)] − 7[(4)(7) − (5)(3)] + 6[(4)(8) − (9)(3)] = 3[63 − 40] − 7[28 − 15] + 6[32 − 27] = 3[23] − 7[13] + 6[5] = 69 − 91 + 30 =8 Answer: 8
9. Find the determinant of this matrix. 1 −5 6 𝐴 = [4 −5 −4] 0 5 −6 4 −4 −5 −4 4 −5 𝑑𝑒𝑡𝐴 = 1 ∙ 𝑑𝑒𝑡 [ ] − (−5) ∙ 𝑑𝑒𝑡 [ ] + 6 ∙ 𝑑𝑒𝑡 [ ] 0 −6 5 −6 0 5 = 1[(−5)(−6) − (−4)(5)] + 5[(4)(−6) − (−4)(0)] + 6[(4)(5) − (−5)(0)] = [30 + 20] + 5[−24 − 0] + 6[20 − 0] = [50] + 5[−24] + 6[20] = 50 − 120 + 120 = 50 Answer: 50 11 © MathTutorDVD.com
10. Find the determinant of this matrix. 4 𝐴 = [−3 6 1 𝑑𝑒𝑡𝐴 = 4 ∙ 𝑑𝑒𝑡 [ −1
−1 2 1 −2] −1 −3
−2 −3 −2 −3 ] − (−1) ∙ 𝑑𝑒𝑡 [ ] + 2 ∙ 𝑑𝑒𝑡 [ −3 6 −3 6
1 ] −1
= 4[(1)(−3) − (−2)(−1)] + [(−3)(−3) − (−2)(6)] + 2[(−3)(−1) − (1)(6)] = 4[−3 − 2] + [9 + 12] + 2[3 − 6] = 4[−5)] + [21] + 2[−3] = −20 + 21 − 6 = −5 Answer: −5
11. Find the determinant of this matrix. 6 −6 4 𝐴 = [3 2 −2] 3 5 −1 𝑑𝑒𝑡𝐴 = 6 ∙ 𝑑𝑒𝑡 [
2 −2 3 −2 3 2 ] − (−6) ∙ 𝑑𝑒𝑡 [ ] + 4 ∙ 𝑑𝑒𝑡 [ ] 5 −1 3 −1 3 5
= 6[(2)(−1) − (−2)(5)] + 6[(3)(−1) − (−2)(3)] + 4[(3)(5) − (2)(3)] = 6[−2 + 10] + 6[−3 + 6] + 4[15 − 6] = 6[8] + 6[3] + 4[9] = 48 + 18 + 36 = 102 Answer: 102 12 © MathTutorDVD.com
12. Find the determinant of this matrix. 3 4 2 𝐴 = [ 5 2 4] 2 3 6 𝑑𝑒𝑡𝐴 = 3 ∙ 𝑑𝑒𝑡 [
2 4 5 4 5 2 ] − 4 ∙ 𝑑𝑒𝑡 [ ] + 2 ∙ 𝑑𝑒𝑡 [ ] 3 6 2 6 2 3
= 3[(2)(6) − (4)(3)] − 4[(5)(6) − (4)(2)] + 2[(5)(3) − (2)(2)] = 3[12 − 12] − 4[30 − 8] + 2[15 − 4] = 3[0] − 4[22] + 2[11] = 0 − 88 + 22 = −66 Answer: −66
13. Find the determinant of this matrix. 6 2 5 𝐴 = [ 3 2 2] 2 8 9 𝑑𝑒𝑡𝐴 = 6 ∙ 𝑑𝑒𝑡 [
2 2 3 2 3 2 ] − 2 ∙ 𝑑𝑒𝑡 [ ] + 5 ∙ 𝑑𝑒𝑡 [ ] 8 9 2 9 2 8
= 6[(2)(9) − (2)(8)] − 2[(3)(9) − (2)(2)] + 5[(3)(8) − (2)(2)] = 6[18 − 16] − 2[27 − 4] + 5[24 − 4] = 6[2] − 2[23] + 5[20] = 12 − 46 + 100 = 66 Answer: 66 13 © MathTutorDVD.com
14. Find the determinant of this matrix. −5 4 −3 𝐴 = [−4 6 −4] 7 8 −2 𝑑𝑒𝑡𝐴 = −5 ∙ 𝑑𝑒𝑡 [
6 −4 −4 ] − 4 ∙ 𝑑𝑒𝑡 [ 8 −2 7
−4 −4 6 ] + (−3) ∙ 𝑑𝑒𝑡 [ ] −2 7 8
= −5[(6)(−2) − (−4)(8)] − 4[(−4)(−2) − (−4)(7)] − 3[(−4)(8) − (6)(7)] = −5[−12 + 32] − 4[8 + 28] − 3[−32 − 42] = −5[20] − 4[36] − 3[−74] = −100 − 144 + 222 = −22 Answer: −22
15. Find the determinant of this matrix. 2 5 −9 𝐴 = [7 6 3] 3 −4 8 𝑑𝑒𝑡𝐴 = 2 ∙ 𝑑𝑒𝑡 [
6 3 7 3 7 6 ] − 5 ∙ 𝑑𝑒𝑡 [ ] + (−9) ∙ 𝑑𝑒𝑡 [ ] −4 8 3 8 3 −4
= 2[(6)(8) − (3)(−4)] − 5[(7)(8) − (3)(3)] − 9[(7)(−4) − (6)(3)] = 2[48 + 12] − 5[56 − 9] − 9[−28 − 18] = 2[60] − 5[47] − 9[−46] = 120 − 235 + 414 = 299 Answer: 299 14 © MathTutorDVD.com
1 © MathTutorDVD.com
Algebra 2 – Matrix Algebra Tutor - Worksheet 9 – Matrix Determinants 1. Find the determinant of this matrix. 𝐴=[
5 3 ] −8 2
2. Find the determinant of this matrix. 12 −3 𝐴=[ ] −7 18
3. Find the determinant of this matrix. 𝐴=[
33 −62 ] −11 37
2 © MathTutorDVD.com
4. Find the determinant of this matrix. 𝐴=[
−2 10
−13 ] −23
5. Find the determinant of this matrix. 𝐴=[
3 6 ] 2 7
6. Find the determinant of this matrix. −6 −6 𝐴 = [−4 −3 −5 −6
3 © MathTutorDVD.com
0 4] 6
7. Find the determinant of this matrix. −5 𝐴 = [−2 3
−1 4 2 −3] 4 6
8. Find the determinant of this matrix. 3 7 6 𝐴 = [ 4 9 5] 3 8 7
4 © MathTutorDVD.com
9. Find the determinant of this matrix. 1 −5 6 𝐴 = [4 −5 −4] 0 5 −6
10. Find the determinant of this matrix. 4 𝐴 = [−3 6
−1 2 1 −2] −1 −3
5 © MathTutorDVD.com
11. Find the determinant of this matrix. 6 −6 4 𝐴 = [3 2 −2] 3 5 −1
12. Find the determinant of this matrix. 3 4 2 𝐴 = [ 5 2 4] 2 3 6
6 © MathTutorDVD.com
13. Find the determinant of this matrix. 6 2 5 𝐴 = [ 3 2 2] 2 8 9
14. Find the determinant of this matrix. −5 4 −3 𝐴 = [−4 6 −4] 7 8 −2
7 © MathTutorDVD.com
Answers - Algebra 2 – Matrix Algebra Tutor - Worksheet 9 – Matrix Determinants Use the following formula find the determinant of a 2 by 2 matrix: 𝐴=[
𝑎 𝑐
𝑏 ] , 𝑑𝑒𝑡𝐴 = 𝑎 ∙ 𝑑 − 𝑏 ∙ 𝑐 𝑑
1. Find the determinant of this matrix. 𝐴=[
5 3 ] −8 2
𝑑𝑒𝑡𝐴 = 5 ∙ 2 − 3(−8) = 10 + 24 = 34 Answer: 34
2. Find the determinant of this matrix. 12 −3 𝐴=[ ] −7 18 𝑑𝑒𝑡𝐴 = 12 ∙ 18 − (−3)(−7) = 216 − 21 = 195 Answer: 195
3. Find the determinant of this matrix. 𝐴=[
33 −62 ] −11 37
𝑑𝑒𝑡𝐴 = 33 ∙ 37 − (−11)(−62) = 1221 − 682 = 539 Answer: 539
8 © MathTutorDVD.com
4. Find the determinant of this matrix. 𝐴=[
−2 10
−13 ] −23
𝑑𝑒𝑡𝐴 = (−2)(−23) − (−13)(10) = 46 + 130 = 176 Answer: 176
5. Find the determinant of this matrix. 𝐴=[
3 6 ] 2 7
𝑑𝑒𝑡𝐴 = 3 ∙ 7 − 6 ∙ 2 = 21 − 12 = 9 Answer: 9
Use the following formula to find the determinant of a 3 by 3 matrix: 𝑎 𝐴 = [𝑑 𝑔 𝑒 𝑑𝑒𝑡𝐴 = 𝑎 ∙ 𝑑𝑒𝑡 [ ℎ
𝑏 𝑒 ℎ
𝑐 𝑓] 𝑖
𝑑 𝑓 ] − 𝑏 ∙ 𝑑𝑒𝑡 [ 𝑔 𝑖
𝑓 𝑑 ] + 𝑐 ∙ 𝑑𝑒𝑡 [ 𝑔 𝑖
𝑒 ] ℎ
6. Find the determinant of this matrix. −6 −6 𝐴 = [−4 −3 −5 −6 𝑑𝑒𝑡𝐴 = −6 ∙ 𝑑𝑒𝑡 [
−3 4 −4 ] − (−6) ∙ 𝑑𝑒𝑡 [ −6 6 −5 9
© MathTutorDVD.com
0 4] 6 4 −4 −3 ] + 0 ∙ 𝑑𝑒𝑡 [ ] 6 −5 −6
= −6[(−3)(6) − (−6)(4)] + 6[(−4)(6) − (−5)(4)] + 0[(−4)(−6) − (−5)(−3)] −6[−18 + 24] + 6[−24 + 20] −6[6] + 6[−4] = −36 − 24 = −60 Answer: −60
7. Find the determinant of this matrix. −5 𝐴 = [−2 3 𝑑𝑒𝑡𝐴 = −5 ∙ 𝑑𝑒𝑡 [
−1 4 2 −3] 4 6
2 −3 −2 ] − (−1) ∙ 𝑑𝑒𝑡 [ 4 6 3
−3 −2 2 ] + 4 ∙ 𝑑𝑒𝑡 [ ] 6 3 4
= −5[(2)(6) − (−3)(4)] + [(−2)(6) − (−3)(3)] + 4[(−2)(4) − (2)(3)] = −5[12 + 12] + [−12 + 9] + 4[−8 − 6] = −5[24] + [−3] + 4[−14] = −120 − 3 − 56 = −179 Answer: −179
10 © MathTutorDVD.com
8. Find the determinant of this matrix. 3 7 6 𝐴 = [ 4 9 5] 3 8 7 𝑑𝑒𝑡𝐴 = 3 ∙ 𝑑𝑒𝑡 [
4 9 9 5 4 5 ] − 7 ∙ 𝑑𝑒𝑡 [ ] + 6 ∙ 𝑑𝑒𝑡 [ ] 3 8 8 7 3 7
= 3[(9)(7) − (5)(8)] − 7[(4)(7) − (5)(3)] + 6[(4)(8) − (9)(3)] = 3[63 − 40] − 7[28 − 15] + 6[32 − 27] = 3[23] − 7[13] + 6[5] = 69 − 91 + 30 =8 Answer: 8
9. Find the determinant of this matrix. 1 −5 6 𝐴 = [4 −5 −4] 0 5 −6 4 −4 −5 −4 4 −5 𝑑𝑒𝑡𝐴 = 1 ∙ 𝑑𝑒𝑡 [ ] − (−5) ∙ 𝑑𝑒𝑡 [ ] + 6 ∙ 𝑑𝑒𝑡 [ ] 0 −6 5 −6 0 5 = 1[(−5)(−6) − (−4)(5)] + 5[(4)(−6) − (−4)(0)] + 6[(4)(5) − (−5)(0)] = [30 + 20] + 5[−24 − 0] + 6[20 − 0] = [50] + 5[−24] + 6[20] = 50 − 120 + 120 = 50 Answer: 50 11 © MathTutorDVD.com
10. Find the determinant of this matrix. 4 𝐴 = [−3 6 1 𝑑𝑒𝑡𝐴 = 4 ∙ 𝑑𝑒𝑡 [ −1
−1 2 1 −2] −1 −3
−2 −3 −2 −3 ] − (−1) ∙ 𝑑𝑒𝑡 [ ] + 2 ∙ 𝑑𝑒𝑡 [ −3 6 −3 6
1 ] −1
= 4[(1)(−3) − (−2)(−1)] + [(−3)(−3) − (−2)(6)] + 2[(−3)(−1) − (1)(6)] = 4[−3 − 2] + [9 + 12] + 2[3 − 6] = 4[−5)] + [21] + 2[−3] = −20 + 21 − 6 = −5 Answer: −5
11. Find the determinant of this matrix. 6 −6 4 𝐴 = [3 2 −2] 3 5 −1 𝑑𝑒𝑡𝐴 = 6 ∙ 𝑑𝑒𝑡 [
2 −2 3 −2 3 2 ] − (−6) ∙ 𝑑𝑒𝑡 [ ] + 4 ∙ 𝑑𝑒𝑡 [ ] 5 −1 3 −1 3 5
= 6[(2)(−1) − (−2)(5)] + 6[(3)(−1) − (−2)(3)] + 4[(3)(5) − (2)(3)] = 6[−2 + 10] + 6[−3 + 6] + 4[15 − 6] = 6[8] + 6[3] + 4[9] = 48 + 18 + 36 = 102 Answer: 102 12 © MathTutorDVD.com
12. Find the determinant of this matrix. 3 4 2 𝐴 = [ 5 2 4] 2 3 6 𝑑𝑒𝑡𝐴 = 3 ∙ 𝑑𝑒𝑡 [
2 4 5 4 5 2 ] − 4 ∙ 𝑑𝑒𝑡 [ ] + 2 ∙ 𝑑𝑒𝑡 [ ] 3 6 2 6 2 3
= 3[(2)(6) − (4)(3)] − 4[(5)(6) − (4)(2)] + 2[(5)(3) − (2)(2)] = 3[12 − 12] − 4[30 − 8] + 2[15 − 4] = 3[0] − 4[22] + 2[11] = 0 − 88 + 22 = −66 Answer: −66
13. Find the determinant of this matrix. 6 2 5 𝐴 = [ 3 2 2] 2 8 9 𝑑𝑒𝑡𝐴 = 6 ∙ 𝑑𝑒𝑡 [
2 2 3 2 3 2 ] − 2 ∙ 𝑑𝑒𝑡 [ ] + 5 ∙ 𝑑𝑒𝑡 [ ] 8 9 2 9 2 8
= 6[(2)(9) − (2)(8)] − 2[(3)(9) − (2)(2)] + 5[(3)(8) − (2)(2)] = 6[18 − 16] − 2[27 − 4] + 5[24 − 4] = 6[2] − 2[23] + 5[20] = 12 − 46 + 100 = 66 Answer: 66 13 © MathTutorDVD.com
14. Find the determinant of this matrix. −5 4 −3 𝐴 = [−4 6 −4] 7 8 −2 𝑑𝑒𝑡𝐴 = −5 ∙ 𝑑𝑒𝑡 [
6 −4 −4 ] − 4 ∙ 𝑑𝑒𝑡 [ 8 −2 7
−4 −4 6 ] + (−3) ∙ 𝑑𝑒𝑡 [ ] −2 7 8
= −5[(6)(−2) − (−4)(8)] − 4[(−4)(−2) − (−4)(7)] − 3[(−4)(8) − (6)(7)] = −5[−12 + 32] − 4[8 + 28] − 3[−32 − 42] = −5[20] − 4[36] − 3[−74] = −100 − 144 + 222 = −22 Answer: −22
15. Find the determinant of this matrix. 2 5 −9 𝐴 = [7 6 3] 3 −4 8 𝑑𝑒𝑡𝐴 = 2 ∙ 𝑑𝑒𝑡 [
6 3 7 3 7 6 ] − 5 ∙ 𝑑𝑒𝑡 [ ] + (−9) ∙ 𝑑𝑒𝑡 [ ] −4 8 3 8 3 −4
= 2[(6)(8) − (3)(−4)] − 5[(7)(8) − (3)(3)] − 9[(7)(−4) − (6)(3)] = 2[48 + 12] − 5[56 − 9] − 9[−28 − 18] = 2[60] − 5[47] − 9[−46] = 120 − 235 + 414 = 299 Answer: 299 14 © MathTutorDVD.com
