This document contains a tutorial on linear algebra concepts including: 1) Calculating determinants of matrices with variables and evaluating them. 2) Finding values of t that satisfy matrix equations. 3) Giving an example of an upper triangular matrix and calculating its determinant. 4) Calculating products, inverses and powers of matrices. It also covers solving systems of linear equations using matrix methods like Cramer's rule and inverse matrices. MATLAB is used to verify the solutions.

1. COLLEGE OF ARTS AND SCIENCES
UNIVERSITI UTARA MALAYSIA
SQQM2023 LINEAR ALGEBRA
TUTORIAL 2
1. Find the determinant of each of the following matrices.
4 0 0 a b c
3 − 5   
a)  b) 5 2 0 c)  0 0 0
 2 1 

 2 0 4
  d e
 f

 2d 2e 2 f  a 3d d
 f   e
d)  d e  e) b 3e 
g
 h i  c 3 f
 f

2. Find all values of t for which
t −1 0 1
t +3 −3
a) =0 b) − 2 t + 2 − 1 = 0
2 t−2
0 0 t +1
3. [ ]
Give an example of an upper triangular matrix A = aij of size 3 × 3 .
Show that A = a11 a 22 a33 .
4. Let A and B be 3 × 3 matrices with A = 4 and B = 5 . Find the values of
−1
a) AB b) 3 A c) 2 AB d) A B
 1 3 − 2 1 0 2 
− 2 1 1 ,  
5. Given A =   and B = 3 2 − 5 .
0 3 0 
  2 1 3 
 
a) Verify that AB = A B
b) Is AB = BA ? Justify your answer.
c) Show that if k =2, then kA = k A
3
[ ]
6. Let A = aij is a 3 × 3 matrix, form the general expression for A by expanding
a) along the first column. b) along the second row.

2. 1 2 − 3 
 
7. Let A = 3 1 4  . Find each cofactor below:
 4 3 − 2
 
a) c13 b) c32 c) c 21
4 2 3 − 4
 3 −2 1 5 
8. Let C =   , find the determinant of C.
− 2 0 1 − 3
 
 8 −2 6 4
9. Let A be a 3x3 matrix. Give an example to show that AA-1 = I, where I is an
identity matrix.
10. Consider the following linear system:
− 2x + 3y − z = 1
x + 2y − z = 4
− 2 x − y + z = −3
a) Write the system as a matrix equation AX = B.
b) Find A .
c) Solve the system using Cramer’s rule.
d) Find adj (A).
e) Solve the system using inverse method.
11. Repeat question 10 (b) – (e) for the linear system below:
1 1 1 − 2  x1  − 4
0 2 1
 3   x2   4 
  =  
2 1 − 1 2   x3   5 
    
1 − 1 0 1   x4   4 
12. Verify your answer in 10 and 11 using MATLAB. The print-out from the matlab
should be attached.