mathematics-class-12-unit-03-chapter-03-matrices-l-3-6-7yetb9n257g

<h3 id="success-stylefont-size25px-matrices-l-3-success-1"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-associativity-of-matrix-multiplication-primary"><primary style="font-size:24px"> Associativity of matrix multiplication </primary></h3>
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<ul>
<li>
<p>Matrix multiplication is associative, i.e. for any 3 matrices $A, B$ & $C$ such that $A $ & $B$are compatible for multiplying and B & C are compatible for multiplying, then <br></p>
</li>
<li>
<p>$A \cdot(B \cdot C)= (A \cdot B) \cdot C$</p>
</li>
<li>
<p>Let, $A$ be a $ n \times m $ matrix<br></p>
</li>
<li>
<p>$B$ be a $m \times r$ matrix<Br></p>
</li>
<li>
<p>& $C$ be a $r \times s$ matrix <br><Br></p>
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<li>
<p>(This is possible because of the given hypothesis)</p>
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<footer style = "font-size: 18px"> $\rightarrow$ Matrix lecture-3 $\rightarrow$ <span style = "color:blue"> Associativity of matrix multiplication </span> $\rightarrow$ Associativity of matrix multiplication proof $\rightarrow$ Associativity of matrix multiplication proof contd... </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-2"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-associativity-of-matrix-multiplication-proof-primary"><primary style="font-size:24px"> Associativity of matrix multiplication proof </primary></h3>
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<p>Let, $A=\left[a_{i j}\right], {1\leqslant i \leqslant n} $, $ ~ $ ${1\leqslant j \leqslant m}$</p>
</li>
<li>
<p>$B=\left[b_{i j}\right], {1\leqslant i \leqslant m}$, $ ~ $ ${1\leqslant j \leqslant r}$</p>
</li>
<li>
<p>$C=\left[c_{i j}\right], {1\leqslant i \leqslant r}$, $ ~ $ ${1\leqslant j \leqslant s} $</p>
</li>
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<footer style = "font-size: 18px">Matrix lecture-3 $\rightarrow$ Associativity of matrix multiplication $\rightarrow$ <span style = "color:blue"> Associativity of matrix multiplication proof </span> $\rightarrow$ Associativity of matrix multiplication proof contd... $\rightarrow$ Associativity of matrix multiplication proof contd... </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-3"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-associativity-of-matrix-multiplication-proof-contd-primary"><primary style="font-size:24px"> Associativity of matrix multiplication proof contd… </primary></h3>
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<p>$A \cdot(B \cdot C) =A \cdot\left(\left[b_{i j}\right] \cdot\left[c_{i j}\right]\right)$</p>
</li>
<li>
<p>=$A \cdot\left(\left[\sum_{k=1}^r b_{i k} c_{k j}\right]\right)$</p>
</li>
<li>
<p>$ =\left[a_{i j}\right]\left(\left[\sum_{k=1}^r b_{i k} c_{k j}\right]\right)$</p>
</li>
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<footer style = "font-size: 18px">Associativity of matrix multiplication $\rightarrow$ Associativity of matrix multiplication proof $\rightarrow$ <span style = "color:blue"> Associativity of matrix multiplication proof contd... </span> $\rightarrow$ Associativity of matrix multiplication proof contd... $\rightarrow$ Associativity of matrix multiplication proof contd... </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Associativity of matrix multiplication proof contd... </primary>
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- $=\left[\sum_{t=1}^m a_{i t} d_{t j}\right]$

- $\text { where } d_{t j} \text { is the }(t, j)^{th}$ entry in the matrix $B \cdot C$
 
- $ =\left[\sum_{t=1}^m a_{i t} \sum_{k=1}^r b_{t k} c_{k j}\right]$

- $ =\left[\sum_{t=1}^m \sum_{k=1}^r a_{i t} b_{t k} c_{k j}\right]$

- $(A \cdot B) \cdot C=\left(\left[a_{i j}\right]\left[b_{i j}\right]\right) \cdot C$

- $=\left(\left[\sum_{k=1}^m a_{i k} b_{k j}\right]\right)$

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<footer style = "font-size: 18px">Associativity of matrix multiplication proof $\rightarrow$ Associativity of matrix multiplication proof contd... $\rightarrow$ <span style = "color:blue"> Associativity of matrix multiplication proof contd... </span> $\rightarrow$ Associativity of matrix multiplication proof contd... $\rightarrow$ Non-commutativity of matrix multiplication </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Associativity of matrix multiplication proof contd... </primary>
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- $=\left(\left[\sum_{k=1}^m a_{i k} b_{k j}\right]\right) \cdot\left[c_{i j}\right]$

- $ =\left[\left(\sum_{t=1}^r \sum_{k=1}^m a_{i k} b_{k t}\right) \cdot c_{t j}\right]$

- $=\left[\sum_{t=1}^r \sum_{k=1}^m a_{i k} b_{k t} c_{t j}\right]$

- $=\left[\sum_{k=1}^r \sum_{t=1}^m a_{i t} b_{t k} c_{k j}\right]=A \cdot(B \cdot C) $

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<footer style = "font-size: 18px">Associativity of matrix multiplication proof $\rightarrow$ Associativity of matrix multiplication proof contd... $\rightarrow$ <span style = "color:blue"> Associativity of matrix multiplication proof contd... </span> $\rightarrow$ Non-commutativity of matrix multiplication $\rightarrow$ Non-commutativity of matrix multiplication contd... </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Non-commutativity of matrix multiplication </primary>
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- **Example-1**

-  Let, $A=\left[\begin{array}{ccc}1 & -2 & 3 \\\ -4 & 2 & 5\end{array}\right]$  & 
- $B=\left[\begin{array}{ll}2 &3 \\\ 4 & 5 \\\ 2 & 1\end{array}\right]$

- $A B=\left[\begin{array}{ccc}1 & -2 & 3 \\\ -4 & 2 & 5\end{array}\right]$ $\left[\begin{array}{ll}2 & 3 \\\ 4 & 5 \\\ 2 &1\end{array}\right]$

- $=\left[\begin{array}{cc}2-8+6 & 3-10+3 \\\ -8+8+10 & -12+10+5\end{array}\right]=\left[\begin{array}{cc}0 & -4 \\\ 10 & 3\end{array}\right]$

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<footer style = "font-size: 18px">Associativity of matrix multiplication proof contd... $\rightarrow$ Associativity of matrix multiplication proof contd... $\rightarrow$ <span style = "color:blue"> Non-commutativity of matrix multiplication </span> $\rightarrow$ Non-commutativity of matrix multiplication contd... $\rightarrow$ Non-commutativity of matrix multiplication </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Non-commutativity of matrix multiplication contd... </primary>
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- $\begin{aligned}
& B A=\left[\begin{array}{ll}
 2 & 3 \\\\
 4 & 5 \\\\
 2 & 1
 \end{array}\right]\left[\begin{array}{ccc}
1 & -2 & 2 \\\\
4 & 2 & 5
\end{array}\right] \\\\
& =\left[\begin{array}{llll}
 2 - 12 & -4+6 & 6+15 \\\\
 4 - 20 & -8+10 & 12+25 \\\\
 2 - 4 & -4+2 & 6+5
\end{array}\right] \\\\
& =\left[\begin{array}{ccc}
10 & 2 & 21 \\\\
16 & 2 & 37 \\\\
2 & -2 & 11
\end{array}\right] \end{aligned}$

- $A ~ B$ is a matrix of order $2 \times 2, ~ B ~ A$ is a matrix of order $3 \times 2$
- $ \Rightarrow A B \neq B A \text {. }$

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<footer style = "font-size: 18px">Associativity of matrix multiplication proof contd... $\rightarrow$ Non-commutativity of matrix multiplication $\rightarrow$ <span style = "color:blue"> Non-commutativity of matrix multiplication contd... </span> $\rightarrow$ Non-commutativity of matrix multiplication $\rightarrow$ Non-commutativity of matrix multiplication contd... </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Non-commutativity of matrix multiplication </primary>
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- **Example-2**

- Let, $A=\left[\begin{array}{ll}1 & 2 \\\ 3 & 4\end{array}\right]$  &  $B=\left[\begin{array}{ll}5 & 6 \\\ 7 & 8\end{array}\right]$

- $\begin{aligned} A B & =\left[\begin{array}{ll}1 & 2 \\\ 3 & 4\end{array}\right]\left[\begin{array}{ll}5 & 6 \\\ 7 & 8\end{array}\right] \\\ & =\left[\begin{array}{ll}5+14 & 6+16 \\\ 15+28 & 18+32\end{array}\right] \\\ & =\left[\begin{array}{ll}19 & 22 \\\ 43 & 50\end{array}\right]\end{aligned}$
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<footer style = "font-size: 18px">Non-commutativity of matrix multiplication $\rightarrow$ Non-commutativity of matrix multiplication contd... $\rightarrow$ <span style = "color:blue"> Non-commutativity of matrix multiplication </span> $\rightarrow$ Non-commutativity of matrix multiplication contd... $\rightarrow$ Zero matrix </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Non-commutativity of matrix multiplication contd... </primary>
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- $B A  =\left[\begin{array}{ll}5 & 6 \\\7 & 8\end{array}\right]\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$

- $=\left[\begin{array}{ll}5+18 & 10+24 \\\7+24 & 14+32\end{array}\right]$

- $=\left[\begin{array}{ll}23 & 34 \\\31 & 46\end{array}\right]$

- Thus, $A B=\left[\begin{array}{ll}19 & 32 \\\ 43 & 50\end{array}\right] \neq\left[\begin{array}{ll}23 & 34 \\\ 31 & 46\end{array}\right]=B A$

- $\therefore A B \neq B A$

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<footer style = "font-size: 18px">Non-commutativity of matrix multiplication contd... $\rightarrow$ Non-commutativity of matrix multiplication $\rightarrow$ <span style = "color:blue"> Non-commutativity of matrix multiplication contd... </span> $\rightarrow$ Zero matrix $\rightarrow$ Row reduced echelon form </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Zero matrix </primary>
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- If $\alpha$ & $\beta$ are any two scalars such that $\alpha \cdot \beta=0$, then either $\alpha=0$ or $\beta=0$.

- $A=\left[\begin{array}{cc}
 0 & -1 \\\\
 0 & 2
\end{array}\right]$ and $B=\left[\begin{array}{ll}
 3 & 5 \\\\ 
 0 & 0
\end{array}\right]$

- $A B=\left[\begin{array}{cc}
 0 & -1 \\\\ 
 0 & 2
\end{array}\right]$
- $\left[\begin{array}{ll}
 3 & 5 \\\\
 0 & 0
\end{array}\right]$$=\left[\begin{array}{cc}
 0+0 & 0+0 \\\\
 0+0 & 0+0
\end{array}\right]$

- $=\left[\begin{array}{ll}
 0 & 0 \\\\
 0 & 0
\end{array}\right] \text { the zero matrix. }$

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<footer style = "font-size: 18px">Non-commutativity of matrix multiplication $\rightarrow$ Non-commutativity of matrix multiplication contd... $\rightarrow$ <span style = "color:blue"> Zero matrix </span> $\rightarrow$ Row reduced echelon form $\rightarrow$ Row reduced echelon form contd... </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Row reduced echelon form </primary>
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- Definition :- A matrix is called row reduced echelon if the following properties hold:

- i) Every zero  row is below every non-zero row.<Br>
- ii) The Leading coefficient (first non-zero coefficient) of every row is 1 .<BR>
- iii) A column which contains the leading coefficient of A, now has all the other coefficients equal to zero.

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<footer style = "font-size: 18px">Non-commutativity of matrix multiplication contd... $\rightarrow$ Zero matrix $\rightarrow$ <span style = "color:blue"> Row reduced echelon form </span> $\rightarrow$ Row reduced echelon form contd... $\rightarrow$ Row reduced echelon form example-1 </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-5"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-row-reduced-echelon-form-example-1-primary"><primary style="font-size:24px"> Row reduced echelon form example-1 </primary></h3>
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<ul>
<li>
<ol>
<li>$\left(\begin{array}{lll}1 & 0 & 2 \\  0 & 0 & 0 \\  0 & 1 & 0\end{array}\right)$<br><br></li>
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<li>The second row is above a non-zero row & hence not a row-reduced echelon matrix.</li>
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<footer style = "font-size: 18px">Row reduced echelon form $\rightarrow$ Row reduced echelon form contd... $\rightarrow$ <span style = "color:blue"> Row reduced echelon form example-1 </span> $\rightarrow$ Row reduced echelon form example-2,3 $\rightarrow$ Row reduced echelon form example-4 </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-6"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-row-reduced-echelon-form-example-23-primary"><primary style="font-size:24px"> Row reduced echelon form example-2,3 </primary></h3>
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<ol start="2">
<li>$\left(\begin{array}{lll}\ 1 & 0 & 1 \\ 0 & \ 2 & 0 \\ 0 & 0 & 0\end{array}\right)$</li>
</ol>
</li>
<li>
<p>The first non-zero coefficient in the second row is 2 & hence not a row reduced  echelon matrix.</p>
</li>
<li>
<ol start="3">
<li>$\left(\begin{array}{lll}\ 1 & 1 & 2 \\  0 & \ 1 & 1 \\  0 & 0 & 0\end{array}\right)$</li>
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<p>Hence, this is not a row reduced echelon matrix.</p>
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<footer style = "font-size: 18px">Row reduced echelon form contd... $\rightarrow$ Row reduced echelon form example-1 $\rightarrow$ <span style = "color:blue"> Row reduced echelon form example-2,3 </span> $\rightarrow$ Row reduced echelon form example-4 $\rightarrow$ Row reduced echelon form example-5 </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-7"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-row-reduced-echelon-form-example-4-primary"><primary style="font-size:24px"> Row reduced echelon form example-4 </primary></h3>
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<li>$\left(\begin{array}{lll}\ 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0\end{array}\right)$ $\quad  k_1=2, \quad k_2=1$,  $k_2<k_1$</li>
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</li>
<li>
<p>$\therefore$ This is not row reduced.</p>
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<footer style = "font-size: 18px">Row reduced echelon form example-1 $\rightarrow$ Row reduced echelon form example-2,3 $\rightarrow$ <span style = "color:blue"> Row reduced echelon form example-4 </span> $\rightarrow$ Row reduced echelon form example-5 $\rightarrow$ Row elementary operations scalar multiplication </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Row reduced echelon form example-5 </primary>
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- 5) $\left(\begin{array}{lll}\ 1 & 0 & 2 \\\ 0 & 1 & 3 \\\ 0 & 0 & 0\end{array}\right)$

* The zero row, the third row is below all the non-zero rows.
* The leading coefficients in both $1^{\text {st }}$ &  2nd rows are just 1.
* All the other elements in a column containing a leading coefficient are zero

- $k_1=1 \quad k_2=2 \quad k_1<k_2$

- Thus, this matrix is a row reduced echelon matrix.

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<footer style = "font-size: 18px">Row reduced echelon form example-2,3 $\rightarrow$ Row reduced echelon form example-4 $\rightarrow$ <span style = "color:blue"> Row reduced echelon form example-5 </span> $\rightarrow$ Row elementary operations scalar multiplication $\rightarrow$ Row elementary operations row interchange </footer>

### <Success style="font-size:25px"> Matrices L-3 </Success>
### <primary style="font-size:24px"> Row elementary operations scalar multiplication </primary>
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- Given a matrix $A$, is there any procedure to convert it into a row reduced echelon matrix?

- Row elementary operations.

- 1) Multiplying the $i^{\text {th }}$ row by a non-zero scalar

- say $\lambda . \quad R_i \rightarrow \lambda R_i$.

- Example: $\left(\begin{array}{lll}1 & 2 & 3 \\\ 4 & 5 & 6\end{array}\right) R_1 \rightarrow 2 R_1\left(\begin{array}{lll}2 & 4 & 6 \\\ 4 & 5 & 6\end{array}\right)$.

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<footer style = "font-size: 18px">Row reduced echelon form example-4 $\rightarrow$ Row reduced echelon form example-5 $\rightarrow$ <span style = "color:blue"> Row elementary operations scalar multiplication </span> $\rightarrow$ Row elementary operations row interchange $\rightarrow$ Row elementary operations replacement by operations </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-8"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-row-elementary-operations-row-interchange-primary"><primary style="font-size:24px"> Row elementary operations row interchange </primary></h3>
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<li>
<ol start="2">
<li>Interchanging $i^{\text {th }}$ row & $j^{\text {th }}$ now.</li>
</ol>
</li>
<li>
<p>$R_i \longleftrightarrow R_j$</p>
</li>
<li>
<p>Eg. $\left(\begin{array}{lll}0 & 1 & 2 \\ 1 & 0 & 3 \\ 0 & 0 & 0\end{array}\right) R_1 \longleftrightarrow R_2\left(\begin{array}{lll}1 & 0 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0\end{array}\right)$.</p>
</li>
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<footer style = "font-size: 18px">Row reduced echelon form example-5 $\rightarrow$ Row elementary operations scalar multiplication $\rightarrow$ <span style = "color:blue"> Row elementary operations row interchange </span> $\rightarrow$ Row elementary operations replacement by operations $\rightarrow$ Procedure to obtain a row reduced echelon matrix from a given matrix </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-9"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
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<ul>
<li>
<ol start="3">
<li>Replace the $i^{\text {th }}$ row by the sum of $i^{\text {th }}$ row & $\mu$-multiple of $j^{\text {th }}$ now.</li>
</ol>
</li>
<li>
<p>$R_i \longrightarrow R_i+\mu R_j \text {. }$</p>
</li>
<li>
<p>Example:  $\left(\begin{array}{llll}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 3\end{array}\right) R_1 \rightarrow R_1+2 R_2\left(\begin{array}{llll}1 & 2 & 2 & 6 \\ 0 & 0 & 1 & 3\end{array}\right)$</p>
</li>
</ul>
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<footer style = "font-size: 18px">Row elementary operations scalar multiplication $\rightarrow$ Row elementary operations row interchange $\rightarrow$ <span style = "color:blue"> Row elementary operations replacement by operations </span> $\rightarrow$ Procedure to obtain a row reduced echelon matrix from a given matrix $\rightarrow$ Procedure to obtain a row reduced echelon matrix from a given matrix contd... </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-10"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-procedure-to-obtain-a-row-reduced-echelon-matrix-from-a-given-matrix-primary"><primary style="font-size:24px"> Procedure to obtain a row reduced echelon matrix from a given matrix </primary></h3>
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<ul>
<li>
<p>Step1: Apply interchange of rows to push down the zero rows to the end of the matrix.</p>
</li>
<li>
<p>Step2: Find the first non-zero column (from left) (let us suppose that it is $k_1$)</p>
</li>
<li>
<p>Step3: Again apply interchange of rows to push up a row where leading non-zero coefficient occurs in first non- zero colum to the first row.</p>
</li>
<li>
<p>Divide the first row by the leading non-zero</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Row elementary operations row interchange $\rightarrow$ Row elementary operations replacement by operations $\rightarrow$ <span style = "color:blue"> Procedure to obtain a row reduced echelon matrix from a given matrix </span> $\rightarrow$ Procedure to obtain a row reduced echelon matrix from a given matrix contd... $\rightarrow$ Example to obtain row reduced echelon form </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-11"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
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<li>
<p>coefficient, so that the leading non-zero coefficient becomes 1.</p>
</li>
<li>
<p>Step 4. Next apply $R_i \longmapsto R_i+\mu R_{1}$ for suitable values of $i$ & $\mu$ so that the first non-zero column has non-zero coefficients only in the first row.</p>
</li>
<li>
<p>Step 5: Repeat steps 2 to 4 for the submatrix obtained by deleting the $1^{\text {st }}$ row & $1^{\text {nd }}$ column until all the non-zero rows are exhausted.</p>
</li>
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<footer style = "font-size: 18px">Row elementary operations replacement by operations $\rightarrow$ Procedure to obtain a row reduced echelon matrix from a given matrix $\rightarrow$ <span style = "color:blue"> Procedure to obtain a row reduced echelon matrix from a given matrix contd... </span> $\rightarrow$ Example to obtain row reduced echelon form $\rightarrow$ Example to obtain row reduced echelon form  </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-12"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-example-to-obtain-row-reduced-echelon-form-primary"><primary style="font-size:24px"> Example to obtain row reduced echelon form </primary></h3>
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<ul>
<li>
<p>Let, $\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right)$</p>
</li>
<li>
<p>$ R_2 \rightarrow R_2+(-1) R_1\\ R_3 \rightarrow R_3+(-1) R_1 \left(\begin{array}{lll}1 & 1 & 2 \\ 0 & 1 & -1 \\ 0 & 1 & 1\end{array}\right)$</p>
</li>
<li>
<p>$ R_1 \longrightarrow R_1+(-1) R_2\\ R_3 \longrightarrow R_3+(-1) R_2 \left(\begin{array}{ccc}1 & 0 & 3 \\ 0 & 1 & -1 \\ 0 & 0 & 2
\end{array}\right)$</p>
</li>
</ul>
</div>
</main>
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<footer style = "font-size: 18px">Procedure to obtain a row reduced echelon matrix from a given matrix $\rightarrow$ Procedure to obtain a row reduced echelon matrix from a given matrix contd... $\rightarrow$ <span style = "color:blue"> Example to obtain row reduced echelon form </span> $\rightarrow$ Example to obtain row reduced echelon form $\rightarrow$ Example to obtain row reduced echelon form  </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-13"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
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<main>
<div style='float: left; width:99%;text-allign:left;font-size: 30px'>
<ul>
<li>$R_3 \longrightarrow\frac{1}{2}R_3\left(\begin{array}{ccc}1 & 0 & 3 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)$</li>
</ul>
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<footer style = "font-size: 18px">Procedure to obtain a row reduced echelon matrix from a given matrix contd... $\rightarrow$ Example to obtain row reduced echelon form $\rightarrow$ <span style = "color:blue"> Example to obtain row reduced echelon form </span> $\rightarrow$ Example to obtain row reduced echelon form $\rightarrow$ Thank You </footer>

<h3 id="success-stylefont-size25px-matrices-l-3-success-14"><Success style="font-size:25px"> Matrices L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-example-to-obtain-row-reduced-echelon-form-primary-2"><primary style="font-size:24px"> Example to obtain row reduced echelon form </primary></h3>
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<main>
<div style='float: left; width:99%; font-size: 30px'>
<ul>
<li>
<p>$ R_1 \rightarrow R_1+(-3) R_3$</p>
</li>
<li>
<p>$ R_2 \rightarrow R_2+R_3$</p>
</li>
<li>
<p>$\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$</p>
</li>
<li>
<p>Thus, the row reduced echelon matrix obtained after applying the procedure to the matrix $\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 1 \\  1 & 2 & 3\end{array}\right)$ is just the identity matrix.</p>
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<footer style = "font-size: 18px">Example to obtain row reduced echelon form $\rightarrow$ Example to obtain row reduced echelon form $\rightarrow$ <span style = "color:blue"> Example to obtain row reduced echelon form </span> $\rightarrow$ Thank You $\rightarrow$  </footer>