MATRICES• Many times we have to encounter the situations

of solving system of linear equations in several variables.

• For example.• 2x + y+ 2z – 12u + v + 2w = 1• x + 2y – z + 6u + 2v – w = 2• 5x + 4y + 3z + 45u + 4v+ 3w = 4• 3x + 20y – 2z – 6u + 8v – 4 w = 2• 20x + y – 2z – 2u + v + 2w = 7 • 2x - 3y + 20z + 5u + 4v – 5w = 19

QUESTIONS :• 1) Whether the solution exists or not ?• 2) Whether there exist(s) ,• (a) Only one solution (UNIQUE)?• (b) More than one solutions or Infinite

solutions?• (c) No solution? and• (d) Can we have a simple method to obtain

the solution (s) ? • Cramer’s Rule and Matrix Inversion could not

answer all the questions (a) (b) (c) (d).

Rank of a matrix• Rank of a matrix helps the methods, which Answer (a)

(b) (c) (d) in a far better way than Cramer’s Rule and Matrix Inversion.

• Sub-matrix of a matrix A: The matrix, obtained by deleting some rows or columns or both of a matrix A.

• Let , then , 1 2 3 02 1 2 3

0 2 1 31 2 6 4

A

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

−−

−−

1 2 3 02 1 2 3

0 2 1 3

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−−

−

1 2 32 1 2

0 2 11 2 6

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−−

−1 2 02 1 3

0 2 3

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−−

are submatrices of A obtained by deleting third row, third column, second column and fourth row, respectively.

Rank of a matrix• Minors of order ‘r’ of a matrix A: The determinant of

any r×r (Square) sub-matrix of m ×n matrix A.

• Let , then ,1 2 32 1 2

0 2 1

−−

1 2 3 02 1 2 3

0 2 1 31 2 3 4

A

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥−⎣ ⎦

1 2 32 1 32 3 4

−−

are the minors of order 3

and1 22 1−

2 3;

1 2−

are the minors of order 2

Rank of a matrixDefinition: Rank of a matrix A is “r” if,

• i) It has at least one non-zero minor of order “r” and

• ii) All the minors of order higher than “r” are zeroes.

• Notation: If Rank of a matrix A is r then it is denoted as ρ(A) = r

• As a consequence of condition (ii), every minor of order greater than r will be zero.

• In short, we say that the rank of a matrix is the largest order of a non-zero minor of the matrix.

Rank of a matrix• If A is a null matrix then ρ(A) = 0.

• If A is not a null matrix then ρ(A) ≥ 1.

• If A is a m ×n matrix, then ρ(A) ≤ min (m, n).

• If A is a square matrix of order n, then ρ(A) = n iff |A| ≠ 0.

• ρ(A) = ρ(AT) .

• If A has a non-zero minor of order ‘r’ then ρ(A) ≥r.

• If all the minors of order ‘r+1’ of A are 0, then ρ(A) ≤r.

Example(s)• Find the rank of each of the following

(a)

(b)

(c)

1 2 34 5 32 4 1

A⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

2 3 43 1 21 2 2

B⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

2 3 44 6 86 9 12

C⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥− − −⎣ ⎦

Solution:

(a) The rank of the matrix is 3.

1 2 34 5 32 4 1

A⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

142354321

=|A| )1016(3)64(2)125(1 −+−−−=

1847 ++−= 015 ≠=

(b) Rank of is 2.2 3 43 1 21 2 2

B⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

•As

Solution:• as , and there is a non-zero minor of

order 2, namely,

(c) Finally for the rank of the matrix

• We see , its rank ≠ 3.• Also every minor of order 2 of C is also zero.• Its rank ≠ 2• As the matrix is a non-zero matrix, so its rank is 1.

2 3 44 6 86 9 12

C =− − −

| | 0C =

| | 0B =

071332

≠−=

Elementary Transformations• The following operations w.r.t. a matrix are known as

elementary transformations.

• Interchange of any two ROWS, indicated by Rij or byRi Rj

• Multiplication of elements of any ROW by a non-zero real number, indicated by Ri →→kRi

• Addition of the constant multiple of jth ROW to ith ROW indicated by Ri →→ Ri + kRj

• Similar COLUMN transformation are denoted by

• Cij , kCi, Ci + kCj

↔