mathematics-class-12-unit-03-chapter-01-matrices-l-1-6-i1f-19adbns-html

### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Matrix-definition </primary>
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- Suppose, there are two companies A and B and each company produces, say, three items I, II & III.

- Suppose the company A produces  70, 80 & 90 kgs of items I, II & III respectively. Similarly, the company B produces 90, 50, & 100 kgs of items I, II & III respectively.

|    | I  | II | III |
|----|----|----|-----|
| A  | 70 | 80 | 90  |
| B  | 90 | 50 | 100 |

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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Matrix-definition </primary>
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- Definition:  A matrix is a rectangular array.

- The elements of the array are called as the entries of the underlying matrix.

- $\left[\begin{array}{ccccc}a_{11} & a_{12} & a_{13} & \cdots & a_{1 n} \\\ a_{21} & a_{22} & a_{23} & \cdots & a_{2 n} \\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\ a_{n 1} & a_{n 2} & a_{n 2} & \cdots & a_{nn}\end{array}\right]$

- There are totally m $\times$ n elements or entries in this matrix.

- There are m rows  and n columns in a matrix.

- The order of a matrix with m rows and n columns is m $\times$ n

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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- Example-1 :   $\quad A=\left[\begin{array}{lll}70 & 80 & 90 \\\\ 90 & 50 & 100\end{array}\right]$
	
- $A$ has got 2 rows and 3 columns.

- $ \therefore$ The order of the matrix $A$ is $2 \times 3$.

- Example-2 : $\quad A=\left[\begin{array}{lll}1 & 2 & 3 \\\ 4 & 5 & 6 \\\ 7 & 8 & 9\end{array}\right]$

- A has got 3 rows and 3 columns

- $\therefore\quad$ order of $A$ is $ ~3 \times 3$.

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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- <ins>**Problem-1**</ins>

- Suppose that $A$ is a real matrix whose order is $3 \times 2$. Find the entries if the entries are given, by the formula $a_{i j}=\frac{|i-j|}{2}$.

- **<ins>Solution</ins>**

- Given,  $A ~ ~ 3 \times 2$ matrix.

- $\therefore\quad A$ has 3 rows and 2 columns.

- $\therefore \quad A$ = <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <msub> <mi>a</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mn>13</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mn>21</mn> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mn>22</mn> </msub> </mtd> <mtd> <msub> <mi>c</mi> <mn>23</mn> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo>=</mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>

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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Kinds of matrices-row matrix </primary>
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- **<ins>Row matrix</ins>** :

- A matrix with just one row is called a row matrix.
- A row matrix will have the order $1\times n$.

- Examples:

- i.) <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> </mstyle> </math> - row matrix $\quad$
- order 1$\times$ 3

- ii.) <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msqrt> <mn>2</mn> </msqrt> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> </mstyle> </math> - row matrix $\quad$
- order 1$\times$ 3

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<h3 id="success-stylefont-size25px-matrices-l-1-success-1"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-kinds-of-matrices-column-matrix-primary"><primary style="font-size:24px"> Kinds of matrices-column matrix </primary></h3>
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<p><strong><ins>Column matrix</ins></strong> :</p>
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<p>A matrix with just one column is called a column matrix.</p>
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<p>The order of a column matrix will be n $\times$ 1</p>
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<p>Examples :</p>
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<p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mtable> <mtr> <mtd> <msqrt> <mn>2</mn> </msqrt> </mtd> </mtr> <mtr> <mtd> <msqrt> <mn>3</mn> </msqrt> </mtd> </mtr> <mtr> <mtd> <msqrt> <mn>5</mn> </msqrt> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math> - column matrix $\quad$ order  $\quad -3\times 1$</p>
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<p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math> - column matrix $\quad$ order  $\quad -3\times 1$</p>
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<h3 id="success-stylefont-size25px-matrices-l-1-success-2"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-kinds-of-matrices-square-matrix-primary"><primary style="font-size:24px"> Kinds of matrices-square matrix </primary></h3>
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<p><ins><strong>Square matrix</strong></ins> :</p>
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<p>A square matrix is a matrix in which the number of rows is equal to the number of columns.</p>
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<p>Examples:</p>
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<p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mn>70</mn> </mtd> <mtd> <mn>80</mn> </mtd> <mtd> <mn>90</mn> </mtd> </mtr> <mtr> <mtd> <mn>90</mn> </mtd> <mtd> <mn>50</mn> </mtd> <mtd> <mn>100</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></p>
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<p>order 2 $\times$ 3 $\quad\therefore ~ $ This is not a square matrix.</p>
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<p>(ii)  <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>27</mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <mn>64</mn> </mfrac> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></p>
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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- Number of rows = 3, $\quad$ Number of columns = 3

- $\therefore\quad$ This is a square  matrix.

- (iii) <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mo></mo> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>

- Number of rows = number of columns = 2

- $\therefore ~ $ This is a square matrix.

- i) A row matrix of order n$\times$1 is a square matrix if and only if n = 1 .

- ii) A column matrix of order n$\times$1 is a square matrix if and only if n = 1.

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<h3 id="success-stylefont-size25px-matrices-l-1-success-3"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-kinds-of-matrices-diagonal-matrix-primary"><primary style="font-size:24px"> Kinds of matrices-diagonal matrix </primary></h3>
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<p><ins><strong>Diagonal matrix</strong></ins> :</p>
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<p>A square matrix is called a diagonal matrix if all the non-diagonal entries are zero.</p>
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<p>$\left[\begin{array}{ccccc}a_{11} & a_{12} & a_{13} & \cdots & a_{1 n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2 n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & a_{n 2} & \cdots & a_{nn}\end{array}\right]$</p>
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<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mstyle> </math> entries are called the diagonal entries.</p>
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<p>Example: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>, this is an example for a diagonal matrix.</p>
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<p>Example: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>, this is an example for a diagonal matrix.</p>
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<p>Example: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>, this is not a  diagonal matrix.</p>
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<p>Example:</p>
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<p>$\left[\begin{array}{lll}
0 & 0 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right]$</p>
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<p>1 is a non-zero entry in the $(2,1)^{th}$ position.</p>
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<p>$\therefore$  This is not a diagonal matrix.</p>
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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Kinds of matrices-scalar matrix </primary>
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- <ins>**Scalar matrix**</ins> :

- A diagonal matrix is called a scalar matrix if all the diagonal entries are obtained  by multiplication by a unique scalar to 1.
	
- Example:  <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math> - scalar matrix.
	
- <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math> - not even a square matrix & hence cannot be a scalar matrix.

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<li>$ \begin{aligned}
I & =\left[\begin{array}{llll}
1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
0 & 0 & \ddots & \vdots\\
\vdots & 0 & \cdots & 1
\end{array}\right]
\end{aligned} $</li>
</ul>
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<p>$
\begin{aligned}
& a_{ij}=\left\{\begin{array}{lc}
0 & \text { if } & i \neq j \\
1 & \text { if } & i=j
\end{array}\right.
\end{aligned}$</p>
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<p>$
\begin{aligned}
& I_{2 \times 2}=\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right] \\
& I_{3 \times 3}=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\end{aligned}
$</p>
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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Kinds of matrices-upper triangular matrix </primary>
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- <ins>Upper triangular matrix</ins> :

- A square matrix in which all the entries that are below the diagonal are zero is called an upper triangular matrix.
	
- Example: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mn>1</mn> <mo></mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>, this is an upper triangular matrix.

- Example: <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mn>2</mn> <mo></mo> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>, this is an upper triangular matrix.

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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Kinds of matrices-upper triangular matrix </primary>
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- Every scalar matrix is an upper triangular matrix. 
	
- Strictly upper triangular matrix. An upper triangular matrix is called strictly upper triangular even the diagonal entries are zero.

- Example : <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>,
- this is an example for a strictly uppertriangular matrix.

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### <Success style="font-size:25px"> Matrices L-1 </Success>
### <primary style="font-size:24px"> Kinds of matrices-upper triangular matrix </primary>
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- Example : <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mn></mn> <mo>.</mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math>

- This is an upper triangular matrix.
	
- 4 is in <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </math>  position and hence this matrix is not strictly upper triangular.

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<h3 id="success-stylefont-size25px-matrices-l-1-success-7"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-kinds-of-matrices-operations-on-matrices-primary"><primary style="font-size:24px"> Kinds of matrices-operations on matrices </primary></h3>
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<p><strong><ins>Operations on matrices:</ins></strong></p>
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<p>Addition of matrices. Two matrices can be added if they are of same order and the <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msup> <mrow> <mo>(</mo> <mi>i</mi> <mi>j</mi> <mo>)</mo> </mrow> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </math> entry of the resulting matrix is obtained by</p>
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<p>adding the $(i, j)^m$ entries of the given two matrices.</p>
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<footer style = "font-size: 18px">Kinds of matrices-upper triangular matrix $\rightarrow$ Kinds of matrices-upper triangular matrix $\rightarrow$ <span style = "color:blue"> Kinds of matrices-operations on matrices </span> $\rightarrow$ Example of addition of matrices $\rightarrow$ Remark </footer>

<h3 id="success-stylefont-size25px-matrices-l-1-success-9"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-remark-primary"><primary style="font-size:24px"> Remark </primary></h3>
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<p>The above said details about real matrices holds without any change to complex matrices.</p>
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<p>A complex matrix is a matrix in which all the entries are complex numbers.</p>
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<p>Example.
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>3</mn> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> <mi>i</mi> </mtd> <mtd> <msqrt> <mn>3</mn> </msqrt> <mo>+</mo> <msqrt> <mn>6</mn> </msqrt> <mi>i</mi> </mtd> <mtd> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <msqrt> <mn>1</mn> </msqrt> <mi>i</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></p>
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<footer style = "font-size: 18px">Kinds of matrices-operations on matrices $\rightarrow$ Example of addition of matrices $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ Note $\rightarrow$ Property A+B=B+A </footer>

<h3 id="success-stylefont-size25px-matrices-l-1-success-10"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-note-primary"><primary style="font-size:24px"> Note </primary></h3>
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<li>Two matrices $A$ & $B$ of same order are said to be equal if each entry of A equals the corresponding entry in $B$. If <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>]</mo> </mrow> </mstyle> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mi>B</mi> <mo>=</mo> <mrow> <mo>[</mo> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>]</mo> </mrow> </mstyle> </math>  then , A = B  if <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </math> for all i and j.</li>
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<footer style = "font-size: 18px">Example of addition of matrices $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Note </span> $\rightarrow$ Property A+B=B+A $\rightarrow$ Property A+(B+C) = (A+B)+C </footer>

<h3 id="success-stylefont-size25px-matrices-l-1-success-11"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-property-abba-primary"><primary style="font-size:24px"> Property A+B=B+A </primary></h3>
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<p>For any two square matrices $A$ & $B$ of same order, $A + B = B+A$</p>
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<p>Proof,</p>
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<p>Let,  $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ where</p>
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<p>$1 \leqslant i_{, j} \leqslant n$.</p>
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<p>$A+B  =\left[a_{i j}\right]+\left[b_{i j}\right] $</p>
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<p>$ =\left[a_{i j}+b_{i j}\right] $</p>
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<p>$ =\left[b_{i j}+a_{i j}\right] $</p>
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<p>$ =\left[b_{i j}\right]+\left[a_{i j}\right] $</p>
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<p>$ =B+A .$</p>
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<footer style = "font-size: 18px">Remark $\rightarrow$ Note $\rightarrow$ <span style = "color:blue"> Property A+B=B+A </span> $\rightarrow$ Property A+(B+C) = (A+B)+C $\rightarrow$ Property A+(B+C) = (A+B)+C </footer>

<h3 id="success-stylefont-size25px-matrices-l-1-success-12"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-property-abc--abc-primary"><primary style="font-size:24px"> Property A+(B+C) = (A+B)+C </primary></h3>
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<p>Thus, $A+B=B+A$.</p>
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<p>This property is called as the commutative property.</p>
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<p>For any three matrices  &  of same order,</p>
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<p>A + ( B + C) = ( A + B ) + C</p>
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<p>Proof, $\quad \text { Let, } A  =\left[a_{i j}\right], $</p>
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<p>$ B  =\left[b_{i j}\right] \ ,  C=\left[c_{i j}\right], 1 \pm i, j \leq n . $</p>
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<p>$ A+(B+C)  =\left[a_{i j}\right]+\left(\left[b_{i j}\right]+\left[c_{i j}\right]\right) $</p>
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<p>$  =\left[a_{i j}\right]+\left(\left[b_{i j}+c_{i j}\right]\right) $</p>
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<footer style = "font-size: 18px">Note $\rightarrow$ Property A+B=B+A $\rightarrow$ <span style = "color:blue"> Property A+(B+C) = (A+B)+C </span> $\rightarrow$ Property A+(B+C) = (A+B)+C $\rightarrow$ Thank you </footer>

<h3 id="success-stylefont-size25px-matrices-l-1-success-13"><Success style="font-size:25px"> Matrices L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-property-abc--abc-primary-1"><primary style="font-size:24px"> Property A+(B+C) = (A+B)+C </primary></h3>
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<p>$  =\left[a_{i j}\right]+\left[b_{i j}+c_{i j}\right]  $</p>
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<p>$ =\left[a_{i j}+\left(b_{i j}+c_{i j}\right)\right] $</p>
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<p>$ \left.=\left[\left(a_{i j}\right)+b_{i j}\right)+c_{i j}\right] $</p>
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<p>$ =\left(\left[a_{i j}+b_{i j}\right]\right)+\left[c_{i j}\right] $</p>
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<p>$ =\left(\left[a_{i j}\right]+\left[b_{i j}\right]\right)+\left[c_{i j}\right] $</p>
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<p>$ =(A+B)+C  $</p>
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<p>Thus, $\quad A+(B+C)=(A+B)+C$.</p>
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<p>This property is called as the associative property</p>
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<footer style = "font-size: 18px"> Property A+B = B+A $\rightarrow$ Property A+(B+C) = (A+B)+C $\rightarrow$ <span style = "color:blue"> Property A+(B+C) = (A+B)+C </span> $\rightarrow$ Thank you $\rightarrow$  </footer>