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.