ADVANCED ENGINERING MATHEMATICS FOR CHEMICAL ENGINEERS
FINAL PAPER WAS HAND WRITTEN
*THERE ARE ERRORS HERE SO REFER TO FINAL OUTPUT
INDIVIDUAL ACTIVITY 1: Matrices and Determinants
I.
What is the row-echelon form of a matrix? What is a leading entry?
Row-echelon form of a matrix is a matrix that must meet a few
requirements to be considered one. First, if there is a row of all zeros, then it is at
the bottom of the matrix and any non-zero rows are always above rows with all
zeroes. Second, the first non-zero element of any row is a one, the leading one.
And lastly, the leading one of any row is to the right of the leading one of the one
above it.
The leading entry is the first non-zero element in each row, 1. Each leading
entry is in a column to the right of the leading entry in the previous row.
II.
(a) What is the augmented matrix of a system? What are leading variables?
An augmented matrix for a system of equations is a matrix in which each
row represents the constants from one equation (both the coefficients and the
constant on the other side of the equal sign) and each column represents all the
coefficients for a single variable.
Leading variables are the variables whose columns in the reduced row
echelon form contain leading 1's.
(b) What are the elementary row operations on an augmented matrix?
Elementary Row Operations are operations that can be performed on a
matrix that will produce a row-equivalent matrix. When a system of linear
equations is converted to an augmented matrix, each equation becomes a row.
There are three(3) elementary row operations on an augmented matrix.
(1) Interchange two rows
(2) Multiply a row by a non-zero constant and
(3) Multiply a row by a non-zero constant and add it to another row, replacing
that row.
(c) How do we solve a system using the augmented matrix?
First step is to make the system of equations into a matrix. Write the
coefficients of the x-terms as the numbers down the first column, y-terms down
on the second column and z-terms on the third column. Draw a vertical line and
write the constants to the right of the line. Then, apply the elementary row
procedures. You may use the Gaussian or Gauss-Jordan Elimination to get the
answers.
(d) Write the augmented matrix of the following system of linear equations.
x  y  2z  3
x  2y  z  5
3 x  y  5z  1
Augmented matrix:
1 1  2 3


1 5
1 2
3  1 5 1
(e) Solve the system in part (d).
1

1
3
2
1
1

1
3
2
1
1
1

0
0

1
1
 2 3
 R1  R 2 
1 5
1

3
5 1
2
1
1
1 5

 2 3
5 1
1 5 R1  R 2  R 2 1
2

1
 2 3
0
1
R3  R 2  R3 
0  4
5 1 3
3

1 2
2
1 5
3

1
3 2  R3  R 2  R30 1
4
4
11 8 
0 0
3
3
3

1 5

3 2
11 8 
3
3

1 5

3 2
23 0 
4

Convert to linear equation;
x  2y  z  5
y  3z  2
23
4
z0
Find the value of z then substitute to the other valid equations;
23 4 z  0
z  o
y  3z  2
y  3( 0)  2
y  2
x  2y  z  5
x  2( 2)  0  5
x 1
x  1; y  2; z  0
III.
Suppose you have used Gaussian elimination to transform the augmented matrix
of a linear system into row-echelon form. How can you tell whether the system has
exactly one solution? no solution? infinitely many solutions?
IV.
What is the reduced row echelon form of a matrix?
A matrix is in reduced row-echelon form when all of the conditions of
row-echelon form are met and all elements above and below a leading one are zero.
1
0


0
V.
0
1
0
0
0

1

(a) How do Gaussian elimination and Gauss-Jordan elimination differ?
Gaussian elimination transforms the matrix into row echelon form and
when converted back into a system of linear equations, back substitution is used
to obtain the answer while Gauss-Jordan elimination transform a matrix into
the reduced row echelon form and no back substitution is needed.
(b) Use Gauss-Jordan elimination to solve the linear system in part II (d).
x  y  2z  3
x  2y  z  5
3 x  y  5z  1
1

1
3
1

3
1
1

0
0
1 1
 2 3
 R 2  R3 
2
1 5
3  1

1 2
 1 5 1
1
1  2 3
 R 2  3R1  R 2 
 1 5 1
0
R3  R1  R3 
0
2
1 5
1
1
2 3 
 1

 4 11  8
R 2  R 2 0
4
0
1
3 2 
1

0
0
1
1
1
1
1 1

0 1

0 0
1 1

0 1
0 0
1 1

0 1
0 0
x  1; y
2.
3.
2 3 

 4 11  8
1
3 2 
1
 2 3

1  11 2
4
1
3 2
1
1 1
3
 2 3



2 R3  R 2  R30 1  11 2
4 

23
2
0
0 0
4 
1 1
 2 3
 2 3
 4


 11 2
R3  R30 1  11 2
4  23
4
0 0
23 0
1 0 
4 
1 1 0 3 
 2 3
R
1

2
R
3

R
1



 11 2
0 1 0 2
4  R 2  11 4 R3  R 2 
0 0 1 0 
1 0 
1 0 0 1 
0 3



0 2 R1  R 2  R10 1 0 2
0 0 1 0 
1 0 
 2; z  0
2
 11
4
3
Matrices A matrix is given.
(a) State the dimension of the matrix.
(b) Is the matrix in row-echelon form?
(c) Is the matrix in reduced row-echelon form?
(d) Write the system of equations for which the given matrix is the augmented
matrix.
VI.
1.
 2 3

5 1
1 5
1
0

1
0

0
2 5 
1 3 
0 8 0
1 5 1 
0 0 0 
0 1 3 4 
 1 1 0 7


 1 2 1 2 
 1 3 6 2
2 1 0 5 


0 0 1 0 
4.
 1 8 6 4 
0 1 3 5 


0 0 2 7 


0
1 1 1
5.
VII.
Matrix Operations
Let
A  2 0 1 
3 
1 / 2

C   2 3 / 2 
 2
1 
1 
 2
E 

 1 / 2 1 
1.
2.
3.
4.
1
B
 2
1
D  0
 2
2 4
1 0 
4
1 
0 
 4 0 2
F   1 1 0 
 7 5 0 
G   5
Carry out the indicated operation, or explain why it cannot be performed.
2C + 3D
5. 5B  2C
GA
6. AG
BC
7. CB
(C + D)E
8. F(2C  D)
1.2C  3D
 12
2C  3D  2  2
 2
1
2C  3D   4
 4
3  1 4 

3   30

1
2
 

1  2 0 
6 3 12 
2  0  3
2 6 0 
4 18 
2C  3D  4  1
2 2 
2.GA
G  1X 1
A  1X 3
GA  52 0  1
GA  10 0  5
3.BC
B  2X 3
C  3X 2
 12
 1 2 4 
BC  
 2

2
1
0

  2

 7 2 1 
BC  

10 9 2 
3
3 
2

1 
4.(C  D ) E
 1 2

(C  D ) E   2
 2

32
(C  D ) E   2
 0
C  D  3X 2
3  1
3   0
2
 
1  2
7 
2
 1 2  
1
2
1  
4 
 2
 1 
 12
0  

 1
1 
E  2X 2
 1 2
(C  D ) E   17 4
 1 2

5 
2

1 
11
2
5. 5B  2C
 12 3 
1 2 4  
5 B  2C  5
 2  2 3 2 

  2 1 0  2 1 


 1 6
 5 10 20  
5 B  2C  
  4 3

 10 5 0  4 2


TOBECONTIN UED...
 1
1 
6. AG
A  1X 3
G  1X 1
DNA
7. CB
 12
CB   2
 2
C  3X 2
3
1 2 4 
3 
2 
  2 1 0

1  
B  2X 3
 11 2
CB    1
  4
2
11
6 
2
 3  8
4
8. F(2C  D)
 4 0 2   1 2

F ( 2C  D )   1 1 0 2  2
 7 5 0 
  2
 4 0 2   1

F ( 2C  D )   1 1 0  4
 7 5 0 
 4
 4 0 2 0 2
F ( 2C  D )   1 1 0 4 4
 7 5 0 6 2
F  3X 3
2C  D  3 X 2
3  1
3   0
2
 
1  2
6  1
3  0
2 2
4 

 1 
0  

4 

 1 
0  

 12 12 
F ( 2C  D )   4
2 
 20 34
VIII.
Matrix Equations
Solve the matrix equation for the unknown matrix X, or show that no solution
exists, where
2 1 
A

 3 2
1. A + 3X = B
2. 2(X  A) = 3B
 1 2
B

 2 4 
 0 1 3
C 

 2 4 0 
3. AX = B
IX.
Determinants and Inverse Matrices
Find the determinant and, if possible, the inverse of the matrix.
1.
3 0 1 
 2 4 0 


4 2 1 
3 0 1
2 - 4 0  3 4 0  0 2 0  2
 2 1
4 1 4





 
4 - 2 1
 3( 4)(1)  0  0  ( 2)( 2)  ( 4)( 4)
 4
 2
 12  12
Det .  0
2.
1
0

0

0
1
0

0

0
0 0 1
2 0 2 
0 3 3

0 0 4
0
0
2
0
0
0
3
0
1
2
2
  10

3


0
4
0
3
0
2
3
000
4

 3 3 
 12 
  0  0
0
4



 12(3)( 4)  0
 12(12)
Det  24
Date of Submission:
March 18, 2021