What Are Matrices? A Beginner's Guide
Get started with matrices with this beginner-friendly guide, covering everything from the start/basics to Matrix Addition, Multiplication, Determinant, Adjoint and Multiplicative Inverse of a Matrix.
This section covers almost all basics like Introduction to Matrices, Types of Matrices, Addition of Matrices and its laws, and Multiplication of Matrices.
Define Matrix:
A matrix is a rectangular array (arrangements) of real numbers enclosed in square brackets.OR
A matrix is an arrangement of real numbers in rows and columns enclosed in square brackets.
Each number in a matrix is called an element or entry of the matrix. Matrices are mostly denoted by capital letters. Like A, B, C etc.
Square Brackets میں Rows اور Columns کے ترتیب کو Matrix کہتے ہیں۔
Matrix میں موجود numbers کو Matrix کے Elements کہتے ہیں۔
Matrixکو عام طور پر Capital Letters سے ظاہر کرتے ہیں۔ جیسے A, B, C
Examples
A=\left[\begin{array}{ll}2 & 3 \\ 0 & 5\end{array}\right]
C=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\end{array}\right]
D=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]
In this example
2, 5,1,3 all are the elements of a matrix D.
2, 5, 1, 3تمام Matrix کے Elementsہے۔
Rows of a Matrix
The rows of a matrix run horizontally.
Columns of a Matrix
The columns of a matrix run vertically.
Example
A=\left[\begin{array}{ll}2 & 3 \\ 0 & 5\end{array}\right]
Here 2, 3 and 0, 5 are row of Matrix A.
Here 2, 0 and 3, 5 are the columns of Matrix A.
In this example 3, 5, 2 and 0, 9, 8 are the rows of a matrix A.
The rows of a matrix run horizontally.
Columns of a Matrix
The columns of a matrix run vertically.
Example
A=\left[\begin{array}{ll}2 & 3 \\ 0 & 5\end{array}\right]
Here 2, 3 and 0, 5 are row of Matrix A.
2, 3اور0,5ہمارے ساتھ Matrix کے Rows ہیں۔
Here 2, 0 and 3, 5 are the columns of Matrix A.
0, 2اور3,5ہمارے ساتھ Matrix کے Rows ہیں۔
A=\left[\begin{array}{lll} 3 & 5 & 2 \\ 0 & 9 & 8 \end{array}\right] \\In this example 3, 5, 2 and 0, 9, 8 are the rows of a matrix A.
3, 5, 2 اور0, 9, 8ہمارے ساتھ Matrix کے Rows ہیں۔
3, 0 , 5, 9 and 2, 8 are the columns of matrix A.5, 9 , 3, 0 اور 2, 8ہمارے ساتھ Matrix کے Columns ہیں۔
Order or Dimension of a Matrix
The number of rows and columns that a matrix has is called order of a matrix.
Order \ of \ matrix =m \times n
OR
Order \ of \ matrix = m-by-n
Here “m” represents number of Rows
And “n” represents number of columns
Note:
Order of a matrix is also called dimension or size of a matrix.
D=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]
In this example
As No. of Rows=2 And No. of Columns=2
Example # 2 A=\left[\begin{array}{lll} 3 & 5 & 2 \\ 0 & 9 & 8 \end{array}\right] \\
In this example
As No. of Rows =2
And No. of Columns =3
The number of rows and columns that a matrix has is called order of a matrix.
Rows اور Columns کی تعداد جو Matrix میں ہوتی ہے اسے Order of a matrix کہا جا تا ہے۔
Order of a matrix is represented by:Order \ of \ matrix =m \times n
OR
Order \ of \ matrix = m-by-n
Here “m” represents number of Rows
And “n” represents number of columns
Note:
Order of a matrix is also called dimension or size of a matrix.
Rows کو “m” سے اور Columns کو “n”سے ظاہر کرتے ہیں۔
ExamplesD=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]
In this example
As No. of Rows=2 And No. of Columns=2
Rowsکی تعداد “2”ہے۔
Columns تعداد “2”ہے۔
So \ order \ is \ 2-by-2 \ (OR) \ 2 \times 2Example # 2 A=\left[\begin{array}{lll} 3 & 5 & 2 \\ 0 & 9 & 8 \end{array}\right] \\
In this example
As No. of Rows =2
And No. of Columns =3
Rowsکی تعداد “2”ہے۔
Columns تعداد “3”ہے۔
So order is 2-by-2 (OR) 2×2Equal Matrix
When two matrices of the same order and the corresponding elements are same.
Solution:
A=\left[\begin{array}{cc}2 & -3 \\ u & 0\end{array}\right]
B=\left[\begin{array}{cc}v & -3 \\ 5 & w\end{array}\right]
As A and B are equal. So
\left[\begin{array}{cc}2 & -3 \\ u & 0\end{array}\right]=\left[\begin{array}{cc}v & -3 \\ 5 & w\end{array}\right]
Now compare the corresponding elements
2=v
Or
v=2
u=5
0=w
Or
w=0
When two matrices of the same order and the corresponding elements are same.
جب دو Matrix کے Order اور تمام Elements ایک جیسے ہو ۔
Let \ A=\left[\begin{array}{cc}2 & -3 \\ u & 0\end{array}\right] \ and \ B=\left[\begin{array}{cc}v & -3 \\ 5 & w\end{array}\right] , for what values of u, \ v, \ and \ w are when A and B equal.Solution:
A=\left[\begin{array}{cc}2 & -3 \\ u & 0\end{array}\right]
B=\left[\begin{array}{cc}v & -3 \\ 5 & w\end{array}\right]
As A and B are equal. So
\left[\begin{array}{cc}2 & -3 \\ u & 0\end{array}\right]=\left[\begin{array}{cc}v & -3 \\ 5 & w\end{array}\right]
Now compare the corresponding elements
2=v
Or
v=2
u=5
0=w
Or
w=0
1. A ____________ is a rectangular array of real numbers enclosed in square brackets.
O Algebra
O Real number
O Matrices
O None
Answer: Matrices
2. Each number in a matrix is called ____________ of the matrix.
O Row
O Entry
O Element
O Both b & c
Answer: Both b & c
3. Matrices are mostly denoted by ____________ letter.
O Small
O Capital
O Both a & b
O None of these
Answer: Capital
4. The ____________ of a matrix run horizontally.
O Row
O Column
O Determinant
O None of these
Answer: Row
5. In \left[\begin{array}{ll}2 & 5 \\ I & 3\end{array}\right] 2,5, I, 3 all are the ____________ of a matrix.
O Row
O Column
O Elements
O None of these
Answer: Elements
6. In \left[\begin{array}{lll}1 & 2 & 3 \\ a & b & c \\ x & y & z \end{array}\right] 2, b \ and \ y are ____________ of a matrix.
O Row
O Column
O Equal
O None of these
Answer: Column
7. In \left[\begin{array}{lll}1 & 2 & 3 \\ a & b & c \\ x & y & z \end{array}\right] , a, b \ and \ c are ____________ of a matrix.
O Row
O Column
O Equal
O None of these
Answer: Row
8. In \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], 1,3 \ and \ 2, 4 are the ____________ of a matrix.
O Rows
O Columns
O Equal
O None of these
Answer: Columns
9. The matrix with m rows and n columns has order ____________
O m \times n
O m-by-n
O both a & b
O None of these
Answer: both a & b
10. A matrix with represents m is ____________
O Row
O Column
O Both a & b
O None of these
Answer: Row
11. A matrix with represents n is ____________
O Row
O Column
O Both a & b
O None of these
Answer: Column
12. Order of matrix can be written as ____________
O Column by row
O Row by Row
O Row by Column
O All of them
Answer: Row by Column
13. The order of matrix \left[\begin{array}{lll}1 & 3 & 5 \end{array}\right] is ____________.
O 2-by-2
O 3-by-3
O 1-by-3
O 3-by-1
Answer: 1-by-3
Explanation:
The Matrix consists of One Row and 3 Columns
14. The order of matrix \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] is ____________.
O 2-by-2
O 3-by-3
O 1-by-3
O 3-by-1
Answer: 2-by-2
Explanation:
The matrix consists of 2 Rows and 2 Columns
15. Order of a matrix is also called _______
O Dimension
O Size
O Both a & b
O None of these
Answer: Both a & b
Explanation:
Order of a matrix is also called Dimension or Size of a matrix.
16. m \times n means _______
O m \ multiply \ n
O order of a matrix
O column of a matrix
O Row of a matrix
Answer: order of a matrix
Explanation:
m \times n does not mean the multiplication.
It shows the order of a matrix having rows and columns.
17. Both \left[\begin{array}{ll}2 & 5 \\ 4 & 3\end{array}\right] and \left[\begin{array}{ll}1+1 & 3+2 \\ 3+1 & 2+1\end{array}\right] are _______
O Equal
O Not equal
O Zero
O None of these
Answer: Equal
Explanation:
\left[\begin{array}{ll}1+1 & 3+2 \\ 3+1 & 2+1\end{array}\right]=\left[\begin{array}{ll}2 & 5 \\ 4 & 3\end{array}\right]
18. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{21} is _______
O 3
O 2
O 4
O 1
Answer: 4
Explanation:
a_{21} means that 2nd Row and First Column. Thus the position of a_{21} is 4.
19. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{22} is _______
O 3
O 2
O 4
O 1
Answer: 1
Explanation:
a_{22} means that 2nd Row and 2nd Column. Thus the position of a_{22} is 1.
20. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{12} is _______
O 3
O 2
O 4
O 1
Answer: 2
Explanation:
a_{12} means that 1st Row and 2nd Column. Thus the position of a_{12} is 2.
21. In \left[\begin{array}{cc}2 & -3 \\ 4 & v\end{array}\right]=\left[\begin{array}{ll}2 & w \\ 6 & 6\end{array}\right] \ then \ w= ________
O 2
O 5
O 6
O -3
Answer: -3
Explanation:
By comparing the corresponding element of w \ which \ is \ -3
22. If \left[\begin{array}{ll}x-1 & 4 \\ y+3 & 7\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -2 & -7\end{array}\right] then \ x= \ ?
O -1
O 0
O 1
O 2
Answer: 1
Explanation:
By comparing the corresponding element of x \ which \ is
x-1=0
x=1
23. If \left[\begin{array}{cc}x-1 & 4 \\ y+3 & -7\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -2 & -7\end{array}\right] \ then\ y= \ ?
O -5
O 5
O 0
O None
Answer: -5
Explanation:
By comparing the corresponding element of y \ which \ is
y+3=-2
y=-2-3
y=-5
O Algebra
O Real number
O Matrices
O None
Answer: Matrices
2. Each number in a matrix is called ____________ of the matrix.
O Row
O Entry
O Element
O Both b & c
Answer: Both b & c
3. Matrices are mostly denoted by ____________ letter.
O Small
O Capital
O Both a & b
O None of these
Answer: Capital
4. The ____________ of a matrix run horizontally.
O Row
O Column
O Determinant
O None of these
Answer: Row
5. In \left[\begin{array}{ll}2 & 5 \\ I & 3\end{array}\right] 2,5, I, 3 all are the ____________ of a matrix.
O Row
O Column
O Elements
O None of these
Answer: Elements
6. In \left[\begin{array}{lll}1 & 2 & 3 \\ a & b & c \\ x & y & z \end{array}\right] 2, b \ and \ y are ____________ of a matrix.
O Row
O Column
O Equal
O None of these
Answer: Column
7. In \left[\begin{array}{lll}1 & 2 & 3 \\ a & b & c \\ x & y & z \end{array}\right] , a, b \ and \ c are ____________ of a matrix.
O Row
O Column
O Equal
O None of these
Answer: Row
8. In \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], 1,3 \ and \ 2, 4 are the ____________ of a matrix.
O Rows
O Columns
O Equal
O None of these
Answer: Columns
9. The matrix with m rows and n columns has order ____________
O m \times n
O m-by-n
O both a & b
O None of these
Answer: both a & b
10. A matrix with represents m is ____________
O Row
O Column
O Both a & b
O None of these
Answer: Row
11. A matrix with represents n is ____________
O Row
O Column
O Both a & b
O None of these
Answer: Column
12. Order of matrix can be written as ____________
O Column by row
O Row by Row
O Row by Column
O All of them
Answer: Row by Column
13. The order of matrix \left[\begin{array}{lll}1 & 3 & 5 \end{array}\right] is ____________.
O 2-by-2
O 3-by-3
O 1-by-3
O 3-by-1
Answer: 1-by-3
Explanation:
The Matrix consists of One Row and 3 Columns
14. The order of matrix \left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] is ____________.
O 2-by-2
O 3-by-3
O 1-by-3
O 3-by-1
Answer: 2-by-2
Explanation:
The matrix consists of 2 Rows and 2 Columns
15. Order of a matrix is also called _______
O Dimension
O Size
O Both a & b
O None of these
Answer: Both a & b
Explanation:
Order of a matrix is also called Dimension or Size of a matrix.
16. m \times n means _______
O m \ multiply \ n
O order of a matrix
O column of a matrix
O Row of a matrix
Answer: order of a matrix
Explanation:
m \times n does not mean the multiplication.
It shows the order of a matrix having rows and columns.
17. Both \left[\begin{array}{ll}2 & 5 \\ 4 & 3\end{array}\right] and \left[\begin{array}{ll}1+1 & 3+2 \\ 3+1 & 2+1\end{array}\right] are _______
O Equal
O Not equal
O Zero
O None of these
Answer: Equal
Explanation:
\left[\begin{array}{ll}1+1 & 3+2 \\ 3+1 & 2+1\end{array}\right]=\left[\begin{array}{ll}2 & 5 \\ 4 & 3\end{array}\right]
18. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{21} is _______
O 3
O 2
O 4
O 1
Answer: 4
Explanation:
a_{21} means that 2nd Row and First Column. Thus the position of a_{21} is 4.
19. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{22} is _______
O 3
O 2
O 4
O 1
Answer: 1
Explanation:
a_{22} means that 2nd Row and 2nd Column. Thus the position of a_{22} is 1.
20. In \left[\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right] \ then \ a_{12} is _______
O 3
O 2
O 4
O 1
Answer: 2
Explanation:
a_{12} means that 1st Row and 2nd Column. Thus the position of a_{12} is 2.
21. In \left[\begin{array}{cc}2 & -3 \\ 4 & v\end{array}\right]=\left[\begin{array}{ll}2 & w \\ 6 & 6\end{array}\right] \ then \ w= ________
O 2
O 5
O 6
O -3
Answer: -3
Explanation:
By comparing the corresponding element of w \ which \ is \ -3
22. If \left[\begin{array}{ll}x-1 & 4 \\ y+3 & 7\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -2 & -7\end{array}\right] then \ x= \ ?
O -1
O 0
O 1
O 2
Answer: 1
Explanation:
By comparing the corresponding element of x \ which \ is
x-1=0
x=1
23. If \left[\begin{array}{cc}x-1 & 4 \\ y+3 & -7\end{array}\right]=\left[\begin{array}{cc}0 & 4 \\ -2 & -7\end{array}\right] \ then\ y= \ ?
O -5
O 5
O 0
O None
Answer: -5
Explanation:
By comparing the corresponding element of y \ which \ is
y+3=-2
y=-2-3
y=-5