 # Matrices

Updated: 07 Mar 2023

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## 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:

##### Define Matrix with explanation and examples
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.
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.

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

2, 5, 1, 3تمام Matrix کے Elementsہے۔

## Rows and Columns of a Matrix

##### Rows and Columns of a Matrix

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.

2, 3اور0,5ہمارے ساتھ Matrix کے Rows ہیں۔

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, 0 , 5, 9 and 2, 8 are the columns of matrix A.

3, 5, 2 اور0, 9, 8ہمارے ساتھ Matrix کے Rows ہیں۔

5, 9 , 3, 0 اور 2, 8ہمارے ساتھ Matrix کے Columns ہیں۔

## Order or Dimension of a Matrix

##### Order or Dimension of a Matrix
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.
Examples
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

Rowsکی تعداد “2”ہے۔

Columns تعداد “2”ہے۔

So \ order \ is \ 2-by-2 \ (OR) \ 2 \times 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

Rowsکی تعداد “2”ہے۔

Columns تعداد “3”ہے۔

So order is 2-by-2 (OR) 2×2

## Equal Matrix

##### Equal Matrix
When two matrices of the same order and the corresponding elements are same.

جب دو Matrix کے Order اور تمام Elements ایک جیسے ہو ۔

## Example of Equal Matrix

##### Example of Equal Matrix
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

## MCQs on Matrix

##### MCQs on Matrix
1. A ____________ is a rectangular array of real numbers enclosed in square brackets.
O Algebra
O Real number
O Matrices
O None

Matrices

2. Each number in a matrix is called ____________ of the matrix.
O Row
O Entry
O Element
O Both b & c

Both b & c

3. Matrices are mostly denoted by ____________ letter.
O Small
O Capital
O Both a & b
O None of these

Capital

4. The ____________ of a matrix run horizontally.
O Row
O Column
O Determinant
O None of these

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

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

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

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

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

both a & b

10. A matrix with represents m is ____________
O Row
O Column
O Both a & b
O None of these

Row

11. A matrix with represents n is ____________
O Row
O Column
O Both a & b
O None of these

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

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

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

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

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

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

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]

\left[\begin{array}{ll}2 & 5 \\ 4 & 3\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

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

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

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

-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

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

-5
Explanation:
By comparing the corresponding element of y \ which \ is
y+3=-2
y=-2-3
y=-5 