Subject: Optional Maths
Definition
A number arranged in rows and columns which enclosed in large brackets or round brackets with a rectangular arrangement is called matrix. It is represented by capital letters . Here, its plural form is matrices .
For eg ;
A = \(\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\) B = \(\begin{bmatrix} 2 & 1 & 3 \\ 6 & 4 & 2 \end{bmatrix}\)
C = \(\begin {bmatrix} 4 & 2 \\ 3 & 2 \\ 1 & 0\end{bmatrix}\)
Rows and colums of a matrix
Matrix is a rectangular arrangement of numbers which contain elements in the form of row and column .Here, a column is usually a vertical line and row is a horizontal line .
eg;
A = \(\begin{bmatrix} 3 & 2 \\ 1 & 5\end{bmatrix}\)
This matrix A has two rows and two columns .
Order of matrix
The number of rows and columns present in a matrix gives the order of a matrix .
eg;
A = \(\begin {bmatrix} 3 & 1 & 2 \\ 5 & 4 & 6 \end {bmatrix}\)
Matrix A has 2 rows and 3 columns So, its order is 2 x 3.And it i read as 2 by 3.
Types of Matrices.
1.Row Matrix
A matrix having only one row is known as a matrix row . It can have any number of columns .
eg; A = \(\begin {bmatrix} 3 & 1 & 2\end{bmatrix}\)
It has only one row .
B = \(\begin{bmatrix} 1 & 2\end {bmatrix}\)
It has only one row .
Column Matrix
2.A matrix having only one column but any number of rows is known as a column matrix .
eg; A =\(\begin{bmatrix} 1 \\2 \\ 3 \end{bmatrix}\) B = \(\begin {bmatrix} 1 \\ 2 \end{bmatrix}\)
Here, A and B both are matrics have only one column .
3.Rectangular Matrix
A matrix having unequal number of rows and columns is known as a rectangular matrix .But,the number of rows and columns should not be equal .
A = \(\begin{bmatrix} 2 & 3 & 4 \\ 7 & 0 & 1 \end{bmatrix}\)
Here matrix A has 2 rows and 3 columns is unequal .So, it is rectangular matrix .
B = \(\begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}\)
4.Zero Matrix
A matrix having each of the elements zero ( 0 ) is known as a null or zero matrix .
O = \(\begin {bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\)
It is a null matrix which is of order 2 x 3.
Square Matrix
A matrix whose number of rows and columns are equal is called a square matrix .
A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) B = \(\begin{bmatrix} 3 & 2 & 1 \\ 4 & 6 & 5 \\ 7 & 9 & 8 \end{bmatrix}\)
Then, In matrix A and B the number of rows and columns both are equal . So, they are squar matrix.
5.Equal Matrices
A pair of matrices having same order and equal corresponding elements are said to be equal matrices .
eg
A = \(\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}\) B = \(\begin{bmatrix} 2 & 3 \\ 4 & 5\end{bmatrix}\)
Here, matrices A and B are said to be equal matrices as their order is same as well as their corresponding elements are equal .
6.Operation on matrices
In this level we shall only see how matrices are added and subtracted . Let's look at the following examples.
A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and B = \(\begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix}\)
A + B = ?
Here, as the order of both the matrices is 2 x 2. we can add them
then, A + B = \(\begin {bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 +8 \end{bmatrix}\) [ We add the corresponding elements ]
=\(\begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix}\)
Again,
A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) C = \(\begin{bmatrix} 5 \\ 6 \end{bmatrix}\)
A + C = ? [ Order is not equal so, no addition is possible ]
= \(\begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix}\) + \(\begin{bmatrix} 5 \\ 6 \end{bmatrix}\)
As the order of these two matrices A and C are not the same we can not add them .
Now,
B  A = ?
so, B  A
= \(\begin{bmatrix} 5  1 & 6  2 \\ 7  3 & 8  4 \end{bmatrix}\)
= \(\begin{bmatrix} 4 & 4 \\ 4 & 4\end{bmatrix}\)
Construct a matrix with the order specified. You can put any elements inside the matrix.
A order 2\(\times\) 3
Solution
A order 2\(\times\)3
It means matrix A should have 2 rows and 3 columns.
A = \(\begin {bmatrix} 1&2&5\\ 2&4&6\\\end {bmatrix}\)
Construct a matrix with the order specified. Put any elements inside the matrix.
C order 3\(\times\)3
Solution
Here, C order 3\(\times\)3
It means matrix C must have 3 rows and 3 columns.
C= \(\begin {bmatrix} 2 & 5& 8\\ 1 & 4 & 7 \\ 3 & 6 & 9\\\end {bmatrix}\)
State the type of matrix given below and also state its order.
A= \(\begin {matrix} 2 &5 \\ 5 & 3 \\ 9 & 7 \\\ \end {matrix} \)
Solution
A= \(\begin {bmatrix}2&5 \\ 5&3 \\ 9&7 \\\ \end {bmatrix} \)
This is rectangular matrix.
It's order is 3 \(\times\) 2.
Determine the order of the following matrix.
P= \(\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}\)
Solution
P= \(\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}\)
Here, Rows present in matrix P= 3
Columns present in matrix p = 3
Therefore, Order of Matrix = 3 \(\times\) 3
Find the value of a, b, c and d if the matrices are equal.
\(\begin {bmatrix} a&5\\6&b\end {bmatrix}\) = \(\begin {bmatrix} 7&c\\d&4\end {bmatrix}\)
Solution
\(\begin {bmatrix} a&5\\6&b\end {bmatrix}\) = \(\begin {bmatrix} 7&c\\d&4\end {bmatrix}\)
Here, As the matrices are equal, their corresponding elements are also equal.
So, a = 7, b = 4, c = 5 and d = 6.
Find the value of x and y if
\(\begin{bmatrix}x5&4\\5&y+3\end {bmatrix}\) = \(\begin {bmatrix}0&2\\4&3\end{bmatrix}\)
Solution
Since the matrices are equal, their corresponding elements are also equal. So,
x  5 = 0
or, x = 5
Again,
y + 3 = 3
y = 0
Hence, x = 5 and y= 0
Find the values of a, b, c and if d if the matrices are equal.
\(\begin{bmatrix}2&3a\\2b & \frac{1}{8}\ \end{bmatrix}\) = \(\begin{bmatrix}4c&6\\ \frac{1}{3}\ &3d\end{bmatrix}\)
Solution
Since the matrices are equal their corresponding elements are also equal.
So, 2 = 4c
or, c = \(\frac{1}{2}\)
Again,
3a = 6
or, a = \(\frac{6}{3}\)
or, a = 2
Similarly,
2b = \(\frac{1}{3}\)
b = \(\frac{1}{3×2}\)
b= \(\frac{1}{6}\)
And,
3d = \(\frac{1}{8}\)
d= \(\frac{1}{3×8}\)
d=\(\frac{1}{24}\)
Therefore, a= 2, b= \(\frac{1}{6}\), c= \(\frac{1}{2}\) and d= \(\frac{1}{24}\).
Find the value of x and y if
\(\begin{bmatrix}5x+1&2\\3&3y2\end{bmatrix}\)=\(\begin{bmatrix}11&2\\3&5\end{bmatrix}\)
Solution
Since the matrices are equal, their corresponding elements are also equal.
So, 5x + 1 = 11
or, 5x = 10
∴ x= 2
Again,
3y  2 = 5
or, 3y = 5 + 2
or, 3y = 3
∴ y = 1
Hence, x = 2 and y = 1.
Find the value of x and y if
\(\begin{bmatrix} x2&0\\0&9\end{bmatrix}\)=\(\begin{bmatrix}3x5&0\\0&y+3\end{bmatrix}\).
Solution
Since the matrices are equal, their corresponding elements are also equal.
So, x2 = 3x5
or, x3x = 5 + 2
or, 2x = 3
x = \(\frac{3}{2}\)
and
or, y+3 = 9
or, y = 9  3
or, y = 6
Hence, x = \(\frac{3}{2}\) and y= 6.
The marks obtained by Renu in class 6 and 7 in three different terminal examinations in mathematics.

1st term  2nd term  Final 
Class 6  40  50  60 
Class 7  70  80  90 
Solution
If we arrange these numbers in a proper arrangement, we can arrange them as
\(\begin{bmatrix}40&50&60\\70&80&90\end{bmatrix}\) or \(\begin{bmatrix}40&50\\60&70\\80&90\end{bmatrix}\)
Where in the first case rows determine class and columns determine terminal exam.
Similarly, in the second case the rows determine the terminal exam and columns determine the class.
State the type of matrix and also state its order.
A = \(\begin{bmatrix}3&5\end{bmatrix}\)
Solution
It is a row matrix.
Its order is 1×2.
Which type of matrix is this? State its order.
O= \(\begin{bmatrix}0&0\end{bmatrix}\)
Solution
It is a null matrix.
Its order is 2×2.
A= \(\begin{bmatrix} 1&2\end{bmatrix} \) and B= \(\begin{bmatrix}2&3\end{bmatrix}\). Find A+B=?
Solution
According to the question,
\(\begin{bmatrix}3&5\end{bmatrix}\)
Find the type of matrix and also state its order.
P= \(\begin{bmatrix}1\\5\\ \end{bmatrix}\)
Solution
It is a column matrix.
Its order is 2×1.
What is Matrix?
A Matrix is a rectangular arrangement of numbers arranged in rows and columns enclosed within large brackets.
© 201920 Kullabs. All Rights Reserved.