Matrices
- 2. A matrix is a rectangular array of real numbers.
Matrix A has 2 horizontal rows and 3 vertical columns.
3 1 −2
A=
7 −1 0.5
Each entry can be identified by its position in the
matrix.
7 is in Row 2 Column 1.
-2 is in Row 1 Column 3.
A matrix with m rows and n columns is of order m × n.
A is of order 2 × 3.
If m = n the matrix is said to be a square matrix of order n.
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- 3. Examples: Find the order of each matrix
2 3 1 0 A has three rows and
A = 4
2 1 4 four columns.
1
1 6 2 The order of A is 3 × 4.
B has one row and five columns.
B = [ 2 5 2 −1 0] The order of B is 1 × 5.
B is called a row matrix.
3 1
C= C is a 2 × 2 square matrix.
6 2
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- 4. An m × n matrix can be written
a11 a12 L a1n
a a22 L a2 n
A = ai j = 21
M
.
M M
am1 am1 am1 amn
Two matrices A = [aij] and B = [bij] are equal if they
have the same order and aij = bij for every i and j.
0.5 9 1
For example, = 2 3 since both matrices
1 7 0.25 7
4
are of order 2 × 2 and all corresponding entries are equal.
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- 5. Matrices
• Write a null matrix or Zero Matrix?
• Write a column matrix?
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- 6. Matrices
• Diagonal Matrix – A square matrix whose
every element other than the diagonal
elements are ZERO is called a diagonal
matrix.
• Diagonal elements – The elements aij are
called diagonal elements when i=j
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- 7. Matrices
• Scalar matrix -A diagonal matrix whose
diagonal elements are equal
• Identity Matrix (Unit matrix) – A diagonal
matrix whose diagonal elements are equal
to one
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- 8. Matrices
• Triangular matrix – A square matrix whose
elements below or above the diagonal are
zero.
• What is an upper triangular matrix?
• What is a lower triangular matrix?
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- 9. To add matrices:
1. Check to see if the matrices have the same order.
2. Add corresponding entries.
Example: Find the sums A + B and B + C.
1 5
A= 2 1 B = 2 0 6 C = 3 −3 0
−1 0 −3 3 2 4
0 6
A has order 3 × 2 and B has order 2 × 3. So they cannot
be added. C has order 2 × 3 and can be added to B.
2 0 6 3 −3 0 5 −3 6
B+C = + 3 2 4 = 2 2 1
−1 0 −3
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- 10. To subtract matrices:
1. Check to see if the matrices have the same order.
2. Subtract corresponding entries.
Example: Find the differences A – B and B – C.
3 7 2 −1 −1 5 1
A= B = 4 −5 C = 2 1 6
2 1
A and B are both of order 2 × 2 and can be subtracted.
3 7 2 −1 1 8
A−B = − 4 −5 = −2 6
2 1
Since B is of order 2 × 2 and C is of order 3 × 2,
they cannot be subtracted.
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- 11. If A = [aij] is an m × n matrix and c is a scalar
(a real number), then the m × n matrix cA = [caij] is the
scalar multiple of A by c. 2 5 −1
3 4 0
Example: Find 2A and –3A for A = .
2 7
2
2(2) 2(5) 2( −1) 4 10 −2
2 A = 2(3) 2(4) 2(0) = 6 8 0
2(2) 2(7) 2(2) 4 14 4
−3(2) −3(5) −3( −1) −6 −15 3
1
− A = −3(3) −3(4) −3(0) = −9 −12 0
3
−3(2) −3(7) −3(2) −6 −21 −6
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- 12. Example: Calculate the value of 3A – 2B + C with
2 −1 5 2 5 2
A = 3 5 B = 1 0 and C = 1 0
4 −2
3 −1
3 −1
2 −1 5 2 5 2
3 A − 2 B + C = 3 3 5 − 2 1 0 + 1 0
4 −2
3 − 1 3 −1
6 −3 10 4 5 2 1 − 5
= 9 15 − 2
0 + 1 0 = 8 15
12 −6 6 −2 3 −1 9 − 5
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- 13. Multiplication of Matrices
• The product AB of two matrices A and B is
defined only when the number of columns
A is same as the number of rows of B.
• A = m x n matrix
• B = n x p matrix.
• The order of AB is m x p.
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- 14. Multiplication of Matrices
• AB may not be equal to BA
• If product AB is defined product BA may
not be defined.
• If A is a square matrix then A can be
multiplied by itself.
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- 15. Transpose of a Matrix
• Let A be a matrix. The matrix obtained by
interchanging the rows and columns is
called the transpose of the matrix A.
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- 16. Symmetric and Skew Symmetric
Matrices
• For a square matrix A if A=AT then it is a
symmetric matrix.
For a square matrix A if A= -AT then it is a
skew symmetric matrix.
Square matrix A + AT symmetric
Square matrix A – AT is skew symmetric
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- 17. • If square matrices AB= BA then A,B are
commutative
• If square matrices AB=-BA the A,B are anti
commutative
• If A2 = A then A is idempotent
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- 18. Determinant of a matrix
• The square matrix A has a uniquely
determined determinant associated with the
matrix.
• The determinant of a product of a matrix is
the product of their determinants.
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- 19. Singular and Non singular
matrices
• A square matrix A is singular if determinant
A is zero.
• A square matrix A is non singular is
determinant A is not equal to Zero.
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- 20. Elementary Operations in matrices.
1. Interchange two rows or columns of a matrix.
2. Multiply a row or column of a matrix by a non
zero constant.
3. Add a multiple of one row or column of a matrix
to another.
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- 21. What is the use of Elementary
operations
• A sequence of elementary row operations
transforms the matrix of a system into the
matrix of another system with the same
solutions as the original system.
• Take matrix A X B = AB
• If we make elementary row operation in AB
then it is equivalent to making the same
operation in A and multiplying it with B.
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- 22. An augmented matrix and a coefficient matrix are
associated with each system of linear equations.
2 x + 3 y − z = 12
For the system
x − 8y = 16
2 3 - 1 12
The augmented matrix is .
1 - 8 0 16
2 3 −1
The coefficient matrix is .
1 −8 0
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- 23. Example: Apply the elementary row operation R1 ↔ R2
to the augmented matrix of the system x + 2 y = 8 .
3x − y = 10
Row Operation Augmented Matrix System
1 2 8 x + 2y = 8
3
-1 10
3x − y = 10
R1 ↔ R2 ↓ ↓
3 -1 10 3x − y = 10
1
2 8 x + 2y = 8
Note that the two systems are equivalent.
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- 24. Example: Apply the elementary row operation 3R2
to the augmented matrix of the system x + 2 y = 8 .
3x − y = 10
Row Operation Augmented Matrix System
1 2 8 x + 2y = 8
3
-1 10
3x − y = 10
3R2 ↓ ↓
1 2 8
9 -3 30
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- 25. Example: Apply the row operation –3R1 + R2
to the augmented matrix of the system x + 2 y = 8 .
3x − y = 10
Row Operation Augmented Matrix System
1 2 8 x + 2y = 8
3
-1 10
3x − y = 10
–3R1 + R2 ↓ ↓
1 2 8 x + 2y = 8
0
-7 - 14 −7 y = −14
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