12M03
Matrices
Matrices
Sachin
Dhoni
12MO3.1
Introduction to Matrices
Learning Objectives
What is Matrix
Types of Matrices
Equality of Matrices
12MO3.1
CV 1
What is Matrix
Row
Column
Matrix Form
Matrix Form
GEETA
RAM
GEETA
RAM
Pen
Pencil
Book
GEETA
RAM
Pen
Pencil
Book
Total no. of Rows
Total no. of Rows
Total no. of Columns
Sol.
Row
Column
Order of the Matrices
Multiplication of total no. of rows and columns
Position of the elements
Total no. of elements
Unique position
Row’s number and Column’s number
Ex.
Ex
Sol.
(iii)
Sol.
Required Matrix=
12MO3.1
CV 2
Types of Matrices
Types of Matrices
Column Matrix
Row Matrix
Square Matrix
Diagonal Matrix
Scalar Matrix
0
Identity Matrix
Zero Matrix
Column Matrix
Matrix which contains only one column.
NO
?
Column Matrix
Column Matrix
,It is a column matrix.
Row Matrix
Matrix which contains only one row.
NO
?
Row Matrix
Row Matrix
,it is a row matrix.
Square Matrix
Matrix which contains equal number of rows and columns.
Square Matrix
Square Matrix
Diagonal Matrix
Diagonal elements
Diagonal element
Square Matrix
Diagonal Matrix
Square matrix which has all non diagonal elements zero.
Diagonal Matrix
Diagonal elements
Scalar Matrix
Diagonal elements
Diagonal matrix
Scalar matrix
Special Case
Diagonal matrix whose all diagonal elements are equal.
Equal
Scalar matrix
Identity Matrix
Diagonal elements
Diagonal matrix
Scalar matrix
Special Case
Equal
Identity matrix
Special Case
Identity matrix
Zero Matrix or Null Matrix
Matrix whose all elements are 0.
Zero Matrix
Zero Matrix
No
?
Q. Mark the correct type.
Matrices
Square
Matrix
Diagonal
Matrix
Scalar
Matrix
Identity
Matrix
12MO3.1
CV 3
Equality of Matrices
Equal Matrices
Two matrices are equal if they have
(i) Same order
(ii) All corresponding elements equal
Q . Find the values of x, y and z from the following equations:� �
Sol.
Both matrices are equal.
All corresponding elements should be equal.
Sol.
Both are equal matrices
All corresponding elements should be equal.
Learning Objectives
Addition of Matrices?
Scalar Multiplication of a Matrix
Difference of Matrices
Multiplication of a Matrices
12MO3.1
CV 3
Addition of Matrices
Addition of Matrices
Room A | ||
| Boys | Girls |
| | |
| | |
| | |
Room B | ||
| Boys | Girls |
| | |
| | |
| | |
Room A + Room B | ||
| Boys | Girls |
| | |
| | |
| | |
Merged
Matrix form
Q .
Sol .
Q.) Compute the following:
Sol.
Addition of corresponding elements
‘’
Commutative Law of Matrix Addition-
Proof.
EQUAL
Proved
Total no. of rows
Total no. of columns
Corresponding
element
Corresponding
row
Corresponding
column
Total no. of rows
Total no. of columns
Associative property of Addition-
Proof.
EQUAL
Proved
12MO3.2
CV 4
Scalar multiplication of Matrices
Ex
Scalar Multiplication of Matrices-
Room A | ||
| Boys | Girls |
| | |
| | |
| | |
Room A | ||
| Boys | Girls |
| | |
| | |
| | |
Doubled
Matrix form=
Multiplication of each element of Matrix by the scalar.
Multiplication of each element of Matrix by the 2.
Scalar
Ex.
Properties Of Scalar Multiplication of Matrices-
Properties Of Scalar Multiplication of Matrices-
Equal
Proved
Negative of a Matrix-
Negating each element of Matrix.
Negative of Matrix A is denoted by –A .
Additive identity existence-
Proof.
Equal
Proved
Additive inverse existence-
Proof.
Zero Matrix
Proved
12MO3.1
CV 5
Difference of Matrices
Difference of Matrices
Subtraction of corresponding elements of matrices
Same order
A-B=?
Sol.
Q.
12MO3.1
CV 6
Multiplication of Matrices
GEETA
RAM
Pencil
Book
Cost
Book
Pencil
Items
Matrix Form
Requirements
Cost
Money needed
GEETA
RAM
Pencil
Book
Cost shop 1
Book
Pencil
Items
Matrix Form
Requirements
Cost
Money needed
Cost shop 2
Requirements
Cost
Money spent
2 Columns
2 Rows
Requirements
Cost
Money spent
2 C
Sol.
Yes
No
Q. Compute the indicated products.
Sol.
Required products
Properties of Matrix Multiplication-
Associative Law-
Proof-
Properties of Matrix Multiplication-
Distributive Law-
Proof-
Properties of Matrix Multiplication-
Multiplicative Identity Law-
Proof-
Identity Matrix
Square Matrix have identity matrix of
same order such that
Square Matrix
Proved
12MO3.1
CV 7
Transpose of a Matrix
Transpose of a Matrix-
Interchanging rows with columns of corresponding matrix or vice versa.
Sol.
Ex.
Q.) Find the transpose of a Matrix given below.
Sol.
Properties of transpose of a matrix-
Proof.
Proved
Equal
Properties of transpose of a matrix-
Proof.
1
0
Proved
Equal
Properties of transpose of a matrix-
Proof.
Proved
Equal
Properties of transpose of a matrix-
Proof.
Equal
Proved
12MO3.1
CV 8
Symmetric and Skew Symmetric Matrix
Symmetric Matrix-
Square Matrix which is equal to its transpose.
Ex.
Equal
Symmetric Matrix
Skew Symmetric Matrix-
Square Matrix whose transpose is equal to its negative .
Ex.
Equal
Skew Symmetric Matrix
Proof
Equal
Proved
Proof
Equal
Proved
Theorem 2-
Any square Matrix can be expressed as sum of Symmetric
and Skew Symmetric Matrix.
Symmetric Matrix
Skew Symmetric Matrix
Q. Find Skew - symmetric matrix of given below
Sol.
12MO3.1
CV 9
Elementary Operation of Matrices
Elementary Operation of Matrices-
Interchanging of any two rows
Ex.
Interchanging of any two columns
Ex.
Multiplication of elements of any row by number
Ex
Any non-zero number
Multiplication of elements of any column by number
Ex
Any non-zero number
S
Addition or subtraction of elements of any row to the corresponding element of any other row multiplied by a number.
Non-zero number
Ex
Addition or subtraction of elements of any column to the corresponding element of any other column multiplied by a number.
Non-zero number
Ex-
12MO3.1
CV 10
Invertible Matrices
Invertible Matrices-
Identity Matrix
Non-Diagonal element
Ex.
Identity Matrix
Inverse of Matrix by Elementary Operation-
Multiplicative Identity
On applying elementary operations on LHS and RHS
LHS
R.H.S.
Inverse of A
LHS
On applying elementary operation if any row or column has
all elements zero then inverse does not exists.
Ex.
Find inverse of matrix
Sol.
Multiplicative inverse
Ex.
Find inverse of matrix
Sol.
Multiplicative inverse
Inverse of a matrix is unique.
Proof .
Identity Matrix
Identity Matrix
Multiplicative Identity
Associative Law
Multiplicative Identity
To Prove
Proved
Inverse of a matrix is unique.
Proof .
Identity Matrix
Identity Matrix
Multiplicative Identity
Associative Law
Multiplicative Identity
To Prove
Proved
A and B are invertible matrices of same order.