CK-12-Pre-Calculus Concepts

8.6. Augmented Matrices http://www.ck12.org Once you have your system represented as a matrix you can solve it using an augmente ...

http://www.ck12.org Chapter 8. Systems and Matrices w+x+z= 11 w+x= 9 x+y= 7 y+z= 5 Solution: While substitution would work in th ...

8.6. Augmented Matrices http://www.ck12.org 3 x+y=− 15 x+ 2 y= 15 Use an augmented matrix to solve the following system. −a+b− ...

http://www.ck12.org Chapter 8. Systems and Matrices − 3 x+y=− 7 −x+ 4 y= 5 3 x+y= 6 − 6 x− 2 y= 10 2 x−y+z= 4 4 x+ 7 y−z ...

8.6. Augmented Matrices http://www.ck12.org 3 x+ 2 y+ 4 z= 21 − 2 x+ 3 y+z=− 11 x+ 2 y− 3 z=− 3 −x+ 2 y− 6 z= 4 8 x+ 5 y+ 3 ...

http://www.ck12.org Chapter 8. Systems and Matrices 8.7 Determinant of Matrices Here you will find the determinants of 2×2 and h ...

8.7. Determinant of Matrices http://www.ck12.org The smaller 2×2 matrices are the entries that remain when the row and column of ...

http://www.ck12.org Chapter 8. Systems and Matrices Example C Find the determinant ofBfrom example B using Sarrus’s Rule. Soluti ...

8.7. Determinant of Matrices http://www.ck12.org Find the determinant of the following matrix. D=   4 8 3 0 1 7 12 5 13   ...

http://www.ck12.org Chapter 8. Systems and Matrices 7.   −1 3 − 4 4 2 1 1 2 5   8.   4 5 8 9 0 1 0 3 − 2   9.   0 7 − ...

8.8. Cramer’s Rule http://www.ck12.org 8.8 Cramer’s Rule Here you will solve systems of equations using Cramer’s Rule. A system ...

http://www.ck12.org Chapter 8. Systems and Matrices [a b c d ][x y ] = [e f ] x= ∣∣ ∣∣e b f d ∣∣ ∣∣ ∣∣ ∣∣a b c d ∣∣ ∣∣ y= ∣∣ ∣∣a ...

8.8. Cramer’s Rule http://www.ck12.org Example A Represent the following system of equations as a matrix equation. y− 13 =− 3 x ...

http://www.ck12.org Chapter 8. Systems and Matrices y= ∣∣ ∣∣∣ ∣∣ ∣ a j c d k f g l i ∣∣ ∣∣∣ ∣∣ ∣ ∣ ∣∣∣ ∣∣ ∣∣ a b c d e f g h i ∣ ...

8.8. Cramer’s Rule http://www.ck12.org x= ∣∣ ∣∣^7212 108 − 12 ∣∣ ∣∣ ∣∣ ∣∣^512 18 − 12 ∣∣ ∣∣ =^725 ··((−−^1212 ))−− 1212 ··^1 ...

http://www.ck12.org Chapter 8. Systems and Matrices 4 x− 2 y=− 20 x− 3 y=− 15 3 x+ 5 y= 33 −x− 2 y=− 13 x+ 4 y= 11 3 ...

8.8. Cramer’s Rule http://www.ck12.org 3 x+ 2 y− 3 z= 7 −x+ 5 y+ 2 z= 29 x+ 2 y+z= 15 Use Cramer’s Rule to solve forxin the fol ...

http://www.ck12.org Chapter 8. Systems and Matrices 6 x+ 3 y− 2 z=− 22 − 4 x− 2 y+ 4 z= 28 3 x+ 3 y+ 2 z= 7 When using Cram ...

8.9. Inverse Matrices http://www.ck12.org 8.9 Inverse Matrices Here you will learn how to find the inverse of a matrix and how t ...

http://www.ck12.org Chapter 8. Systems and Matrices x+ 2 y+ 3 z= 96 x+ 0 y+z= 36 0 x+ 2 y−z=− 12 By writing this system as a mat ...