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Definition: For a 2 2
matrixba
ADefinition: For a 2 2
matrixdc
baA
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.summarizes the whole matrix A.
Definition: aaaDefinition:The determinant
131211
aaa
aaa
AThe determinant of a 3 3 matrix:
333231
232221
aaa
aaaA
is defined by: |A| =333231 aaa
aaa2221
132321
122322
11232221
131211
aa
aaa
aa
aaa
aa
aaaaaa
aaa
323133313332333231
aaaaaaaaa
Let A be an n n matrix,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 Acolumn j from A
Also define C = ( 1)i+jdet(M ) Also define Cij = ( 1)i+jdet(Mij)
to be the (i,j)-cofactor of A. to be the (i,j)-cofactor of A.
Then, the determinant of A can be computed by:
det(A) = j aijCij = ai1Ci1 + ai2Ci2 + + ainCindet(A) = j aijCij = ai1Ci1 + ai2Ci2 + + ainCin
(a cofactor expansion along the ith row)
or by:or by:
det(A) = i aijCij = a1jC1j + a2jC2j + + anjCnj
(a cofactor expansion along the jth column).
the determinant of the matrixFind
the determinant of the matrix
3 1 -4
Find
3 1 -4
A= 2 5 6A= 2 5 6
1 4 81 4 8
using the first row
using the second column
5 6 2 6 2 5( ) 3 1 4A( ) 3 1 4
4 8 1 8 1 4
A
=3(16)-10-4(3)=26
=3(16)-10-4(3)=26
2 6 3 4 3 4( ) 1 5 4A
2 6 3 4 3 4( ) 1 5 4
1 8 1 8 2 6A
1 8 1 8 2 6
=-10+5(28)-4(26) =26
If A has a zero row or column, then |A| = 0.1. If A has a zero row or column, then |A| = 0.
2. If A is upper or lower triangular matrix, then |A| = a11a22 ann.|A| = a11a22 ann.
3. If A is a diagonal matrix, then3. If A is a diagonal matrix, then|A| = a11a22 ann.
4. |In| = 14. |In| = 1
5- If B is obtained by switching two rows (or columns) of A, then |B| = |A|.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|.of A by k, then |B| = k|A|.
7- If B is obtained by adding a multiple of a row (or 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|.|B| = |A|.
8- |A| = |AT|8- |A| = |AT|
9- If two rows (columns) are identical then
|A| = 0|A| = 0
10- |AB| = |A| |B| if A and B are of the same
order.order.
11- |kA| = kn |A|
A mxn matrix can be written asR
A= 1
2.
RR..
mR
Ri=[ai1,ai2, ,ain] row i of AAlso we can write A as A=[C1,C2, ,Cn]Also we can write A as A=[C1,C2, ,Cn]where Cj is column j of A
2 4 82 4 8
3 6 12
1 5 9
A
1 2 4
1 5 9
1 2 41 2 4
= 3 6 3
1
12 02
2 4
2 1 2 4x
1 5 9 1 5 9
6 4 56 4 5
5 4 6B
1 0 1
1 0 12 1 5 4
1 0 1
6 0R R
1 0 1
12 9 3 4 3 2
0 5 4 0 5 4
4 3 2 12 9 3
A
4 3 2 12 9 3
4 3 23 1 3
4 3 2
0 5 4 (4)(5)( 3) 60R R
0 0 3
A row Rs is said to be a linear combination of R ,R , ,R if there exist real numbers
sof R1,R2, ,Rm if there exist real numbers k1,k2, ,km such thatk1,k2, ,km such that
R = k R +k R + +k RRs = k1R1+k2R2+ +kmRm
For the matrix A, defined below, show that R2can be written as a linear combination of the can be written as a linear combination of the rows of A
1 3 2 4
3 5 0 7A
3 5 0 7
2 1 5 2
3 0 1 1
A
R =R -R +2R
3 0 1 1
R2=R4-R3+2R1
For the matrix A, defined below, show that C3 can be written as a linear combination of the columns of A
1 2 31 2 3
2 3 5A 2 3 5
2 2 4
A
3 1 2
2 2 4
C C C3 1 2
If a row (column) of a matrix A can be expressed as a linear combination of the expressed as a linear combination of the other rows (columns) we say that the rows other rows (columns) we say that the rows (columns) of A are linearly dependent
The rows of a matrix A are linearly independent if the only solution ofindependent if the only solution of
k1R1+k2R2+ +kmRm=0k1R1+k2R2+ +kmRm=0is k1=k2= =km=01 2 m
i.e. any row cannot be written as a linear combination of the other rowslinear combination of the other rows
If the rows (columns) of A are linearly dependent then dependent then
Use the determinates properties to show
that (A) = 0
2 1 1
4 1 5
1 2 3 9
A
1 2 3 9
CC C1 2 3
CC C
A square matrix A is invertible if and only if
(A) 0(A) 0
A square matrix A is invertible if and only if its rows (columns) are linearly its rows (columns) are linearly independentindependent
If A is invertible then |A-1| = 1/|A|If A is invertible then |A-1| = 1/|A|Proof:Proof:Since A is invertible, then AA-1=In|AA-1| = |In| = 1|AA-1| = |A| |A-1| =1|AA-1| = |A| |A-1| =1Since |A| 0, then |A-1| = 1/|A|
Let A be an n n square matrix. The following Let A be an n n square matrix. The following statements are all equivalent:
1.
2.
3.3.
4.4.
Find all values of k, for which the following matrix is invertible:matrix is invertible:
k
k
A 22
22
k
kA
22
22
k22
If k=2 then A =0
k1 2C C k k
k
k kA k
k kA k
k k2 k k2 3
Show that x=3 is one of the roots of the equationequation
Show that the matrix
1 2 31 0 1A 1 0 12 4 6
A
is not invertible2R2=R1
A =02R2=R1
A =0
A non-zero matrix A is said to have rank k r(A) = k
if at least one of its k-square minors is if at least one of its k-square minors is different from zero while every (k+1)-square minors, if any, is zero.square minors, if any, is zero.A zero matrix is said to have rank zero.A zero matrix is said to have rank zero.
mxnmxn
An n-square matrix is said to be full rank matrix if r(A) = n.matrix if r(A) = n.
Result:The n-square matrix A is invertible if and only if r(A) = ninvertible if and only if r(A) = n
Find the rank of A = 2 1 1
4 1 5
C =C -C
12 3 9
C2=C1-C3
A = 02 1M33 = 2 1
6 04 1
r(A) =2
Find the rank of A = 1 2 3Find the rank of A = 1 2 3
5 10 15
2 4 6
R3
= 2
R1
2 4 6
R3
= 2
R1
R2 = -5 R1
r(A) = 1r(A) = 1Note that A =0 and all 2x2 minors are zero also.also.
The following operations, called elementary transformations on a matrix do elementary transformations on a matrix do not change either its order or its rank:not change either its order or its rank:
1- Interchanging two rows (columns)1- Interchanging two rows (columns)
2- The multiplication of every element of of row (column) by a nonzero constant k.row (column) by a nonzero constant k.
3- The multiplication of every element of a row (column) by a nonzero constant k and row (column) by a nonzero constant k and adding the result to another row (column).adding the result to another row (column).
Two matrices A and B are called equivalent, A B , if one can be obtained equivalent, A B , if one can be obtained from the other by a sequence of from the other by a sequence of elementary transformations.
Equivalent matrices have the same order and the same rank.and the same rank.
1 2 1
1 2
1 2 1
2 4 3 2
1 2 1
A R R
1 2 1
1 2 1
30 0 5
1 2 11 R R
1 2 1
1 2 1
0 0 5
0 0 5
0 0 0
Show that the following matrix A is equivalent to the identity matrix Iequivalent to the identity matrix I2
2 2A
2 2
1 4A
1
1 11R A1 1 42R A
1 2
1 1
0 3R R A1 2 0 3R R A
2 2
1 11
0 13R A I
0 13
Given an n-square matrix A, the following Given an n-square matrix A, the following statements are equivalent:statements are equivalent:1- A is invertible.2- r(A) = n.2- r(A) = n.3- A In3- A In4- A 05- All rows of A are linearly independent.5- All rows of A are linearly independent.