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Introduction to Matrices

2. • In 1985 Arthur caylay presented the system of matrices called Theory of matrices.
• It was the latest way to solve the systems of linear equations.
• For example x +2y=0
3x+4y=0
• As we know in equation system we deal with coefficients of variables to solve the problems, so Arthur
caylay plot these coefficients in this form.
• We denote a Matrix with capital letters A, B, C and so on.
• The numbers in a matrix is called its entries or elements.
• We define matrix as A collection of numbers in a rectangular array is called Matrix.
• We enclose the elements in [ ] Square brackets or in ( ) parenthesis.
• The elements are arranged in rows and columns. (1 2) is in Row 1 denoted by R1 and (3 4) are in R2.
While 1 and 2 are in Column form denoted by C1 and C2
3 4
• Or we can say that when we write numbers in a horizontal way this is called the row of the matrix and
when we write the numbers in vertical way we called it as columns of the matrix.
• We may have many rows and columns in a matrix.

3. Order of Matrix
• Order of matrix means how many rows and columns are their in a matrix.
• In order of matrix total number of rows are denoted by m and total number of columns
are denoted by n.
• We can say that order of a matrix is no of rows by no of columns such that m x n
(read as m-by-n).
• For example we have a matrix
• Now look at the matrix
a11 means 1st row and 1st column
a12 means 1st row and 2nd column
a13 means 1st row and 3rd column up to a1n means 1st row and columns
Similarly a21 means 2nd row and 1st column
a22 means 2nd row and 2nd column and a23 means 2nd row and 3rd
column up to a2n means 2nd row and n columns.
• We usually denote each element in matrix by amn means a number
Say 5 in 1st row m and 2nd column n.
• The order of matrix help us to perform the mathematical operations.

4. Types of Matrices
Equal Matrices
• Two matrices or more are said to be equal if their corresponding elements and order are the same. Such
that we have matrix A of order 2 x 2 and matrix B 2 x 2
• For example
• So for equal
Matrices we must
have 1. same order
2. same corresponding elements.
• We use equal matrices to find out the unknown variables by comparing
corresponding elements.

5. Row and Column Matrices
• A matrix with only one row such that by order 1 x n is said to be row matrix.
• In row matrix we have no concern with the number of columns n, we only look at the no of row which
should be one called row matrix.
• Fro example
Which have one row and 3 columns such that 1 x 3
• A matrix with only one column such that by order m x 1 is said to be column matrix.
• In column matrix we have no concern with the number of roes m, we only look at the no of column which
should be one called column matrix
• For example
• It is possible a matrix may be at the same be row and column matrix that is 1 x 1 for example [7].

6. Square, Rectangular and Zero Matrices
• A matrix is said to be square matrix if m = n means no of rows should be equal to number of columns.
• For example
Which have 2 rows
And 2 columns
• A matrix is said to be rectangular matrix if m ≠ n means no of rows should not be equal to number of
columns.
• For example
Which have 2 rows
And 3 columns
• A matrix is said to be zero or null matrix whose have elements are zero 0. Irrespective of order of matrix.
• For example
• Remember 0 is a real number which is called the additive identity because whenever we add 0 with any
number it gives the same number as a answer. So we can say a matrix also contain zero matrix.
• A zero matrix may be row, column, square or rectangular matrix.
• It is denoted by o.
• It is used to find out the additive inverse .

7. Diagonal and Scalar Matrices
• For diagonal matrix the following conditions should be there.
1. It should be square matrix
2. It should contains at least 1 non zero element in its diagonal.
3. While other elements should be zero 0.
• A straight line inside a shape that goes from one corner to another is called diagonal.
• For example the line joining A to B is called diagonal
• Diagonal matrix in which its diagonal elements have at least 1 non zero element and other are zero
elements.
• A diagonal matrix which have the same diagonal and non zero elements called scalar matrix.

8. Identity Matrix
• A scalar matrix which have 1 in its diagonal.
• It is denoted by I.
• It is called identity matrix because when we
Multiply this with any matrix it gives the same matrix, the other matrix did not lost his identity.
• Identity matrix is always a square matrix.

9. Addition and Subtraction of Matrices
• Two matrices can be added if matrix A and B have the same order..
• The entries or elements are added with their corresponding entries.
• Two matrices can be subtracted if matrix A and B have the same order ..
• The entries or elements are subtracted with their corresponding entries.
• In subtraction matrices A – B ≠ B – A .

10. Transpose and Negative of Matrix
• Let A be a matrix of order 2-by-3, then transpose of a matrix means interchanging rows by columns and
columns by rows.
• For example
• It is denoted by At
• For a matrix of order
Say 2 by 3 then its
transpose will change the
order by 3 by 2.
• Remember all real numbers have negative called additive inverse i.e 7 has -7. Similarly in matrices we
also have negative matrix.

11. Multiplication of Matrices
Scalar multiplication
• Two or more matrices can be multiplied if no of columns in first matrix equals to no of rows in second matrix.
Such that (no of columns in first matrix) n = m (no of rows in second matrix)
• The 1st row is multiplied by 1st column of other
matrix, then same row is multiplied by 2nd column of
other matrix and their product are added.
• The product we get from multiplication must have
the order of no of rows of 1st matrix and no of columns
of 2nd matrix. From their product we get matrix of order
2 by 2
• Scalar multiplication is the multiplication with the
real number whether positive or negative.