3. a b
Definition: For a 2 2 matrix A
c d
define its determinant by: det(A) = |A| = ad bc
Observe that det(A) is a scalar that in a way
summarizes the whole matrix A.

4. Definition: a11 a12 a13
The determinant A a21 a22 a23
of a 3 3 matrix:
a31 a32 a33
is defined by: |A| =
a11 a12 a13
a22 a23 a21 a23 a21 a22
a21 a22 a23 a11 a12 a13
a32 a33 a31 a33 a31 a32
a31 a32 a33

5. Let A be an n n matrix,
define Mij to be the (i,j)-minor of A,
i.e. the resulting matrix after removing row i and
column j from A
Also define Cij = ( 1)i+jdet(Mij)
to be the (i,j)-cofactor of A.

6. Then, the determinant of A can be computed by:
det(A) = j aijCij = ai1Ci1 + ai2Ci2 + + ainCin
(a cofactor expansion along the ith row)
or by:
det(A) = i aijCij = a1jC1j + a2jC2j + + anjCnj
(a cofactor expansion along the jth column).

7. Find the determinant of the matrix
3 1 -4
A= 2 5 6
1 4 8
using the first row
using the second column

8. 5 6 2 6 2 5
(A ) 3 1 4
4 8 1 8 1 4
= 3 (1 6 )-1 0 -4(3 )= 2 6

9. 2 6 3 4 3 4
(A ) 1 5 4
1 8 1 8 2 6
=-10+5(28)-4(26) =26

10. 1. If A has a zero row or column, then |A| = 0.
2. If A is upper or lower triangular matrix, then
|A| = a11a22 ann.
3. If A is a diagonal matrix, then
|A| = a11a22 ann.
4. |In| = 1

11. 5- If B is obtained by switching two rows (or columns)
of A, then |B| = |A|.
6- If B is obtained by multiplying a row (or a column)
of A by k, then |B| = k|A|.
7- If B is obtained by adding a multiple of a row (or a
column) of A to another row (column), then
|B| = |A|.

12. 8- |A| = |AT|
9- If two rows (columns) are identical then
|A| = 0
10- |AB| = |A| |B| if A and B are of the same
order.
11- |kA| = kn |A|

13. A mxn matrix can be written as
R1
A= R2
.
.
Rm
Ri=[ai1,ai2, ,ain] row i of A
Also we can write A as A=[C1,C2, ,Cn]
where Cj is column j of A

14. 2 4 8
A 3 6 12
1 5 9
1 2 4 1 2 4
=2 3 6 12 2x 3 1 2 4 0
1 5 9 1 5 9

15. 6 4 5
B 5 4 6
1 0 1
1 0 1
R R1
2
5 4 6 0
1 0 1

16. 12 9 3 4 3 2
A 0 5 4 0 5 4
4 3 2 12 9 3
4 3 2
3R 1 R 3
0 5 4 (4)(5)( 3) 60
0 0 3

17. A row Rs is said to be a linear combination
of R1,R2, ,Rm if there exist real numbers
k1,k2, ,km such that
Rs = k1R1+k2R2+ +kmRm

18. For the matrix A, defined below, show that R2
can be written as a linear combination of the
rows of A
1 3 2 4
3 5 0 7
A
2 1 5 2
3 0 1 1
R2=R4-R3+2R1

19. For the matrix A, defined below, show that C3 can
be written as a linear combination of the columns
of A
1 2 3
A 2 3 5
2 2 4
C 3 C1 C 2

20. If a row (column) of a matrix A can be
expressed as a linear combination of the
other rows (columns) we say that the rows
(columns) of A are linearly dependent

21. The rows of a matrix A are linearly
independent if the only solution of
k1R1+k2R2+ +kmRm=0
is k1=k2= =km=0
i.e. any row cannot be written as a
linear combination of the other rows

22. If the rows (columns) of A are linearly
dependent then

23. Use the determinates properties to show
that (A) = 0
2 1 1
A 4 1 5
12 3 9
C 1 C 2 C 3

24. A square matrix A is invertible if and only if
(A) 0

25. A square matrix A is invertible if and only if
its rows (columns) are linearly
independent

26. If A is invertible then |A-1| = 1/|A|
Proof:
Since A is invertible, then AA-1=In
|AA-1| = |In| = 1
|AA-1| = |A| |A-1| =1
Since |A| 0, then
|A-1| = 1/|A|

27. Let A be an n n square matrix. The following
statements are all equivalent:
1.
2.
3.
4.

28. Find all values of k, for which the following
matrix is invertible:
k 2 2
A 2 k 2
2 2 k

29. If k=2 then A =0
k
C1 C 2
k k
k

30. k k
A k
k k
2
k k
2 3

31. Show that x=3 is one of the roots of the
equation

33. Show that the matrix
1 2 3
A 1 0 1
2 4 6
is not invertible
2R2=R1 A =0

34. A non-zero matrix A is said to have rank k
r(A) = k
if at least one of its k-square minors is
different from zero while every (k+1)-
square minors, if any, is zero.
A zero matrix is said to have rank zero.

35. mxn

36. An n-square matrix is said to be full rank
matrix if r(A) = n.
Result:
The n-square matrix A is
invertible if and only if r(A) = n

38. Find the rank of A = 2 1 1
4 1 5
12 3 9
C2=C1-C3
A =0
M33 = 2 1
6 0
4 1
r(A) =2

39. Find the rank of A = 1 2 3
5 10 15
2 4 6
R3 = 2 R1
R2 = -5 R1
r(A) = 1
Note that A =0 and all 2x2 minors are zero
also.

41. The following operations, called
elementary transformations on a matrix do
not change either its order or its rank:
1- Interchanging two rows (columns)
2- The multiplication of every element of of
row (column) by a nonzero constant k.

42. 3- The multiplication of every element of a
row (column) by a nonzero constant k and
adding the result to another row (column).

43. Two matrices A and B are called
equivalent, A B , if one can be obtained
from the other by a sequence of
elementary transformations.

44. Equivalent matrices have the same order
and the same rank.

45. 1 2 1
A 2 4 3 2R1 R 2
1 2 1
1 2 1
0 0 5 R1 R 3
1 2 1
1 2 1
0 0 5
0 0 0

46. Show that the following matrix A is
equivalent to the identity matrix I2
2 2
A
1 4

47. 1 1 1
R1 A
2 1 4
1 1
R1 R 2 A
0 3
1 1 1
R 2 A I 2
3 0 1

48. Given an n-square matrix A, the following
statements are equivalent:
1- A is invertible.
2- r(A) = n.
3- A In
4- A 0
5- All rows of A are linearly independent.