mathematics-class-12-unit-03-chapter-02-matrices-l-2-6-uifm7sib7uq

### <Success style="font-size:25px"> Matrices L-2 </Success>
### <primary style="font-size:24px"> Property $(\alpha+\beta) \cdot A$ </primary>
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<main>

- i) $(\alpha+\beta) \cdot A=\alpha \cdot A+\beta \cdot A$

- Let, $A=\left[a_{i j}\right] then,$

- $(\alpha+\beta) \cdot A  =\left[(\alpha+\beta) \cdot a_{i j}\right]$

- $ =\left[\alpha \cdot a_{i j}+\beta \cdot a_{i j}\right] $

- $ =\left[\alpha \cdot a_{i j}\right]+\left[\beta \cdot a_{i j}\right]$

- $ =\alpha \cdot\left[a_{i j}\right]+\beta \cdot\left[a_{i j}\right]$

- $ =\alpha \cdot A+\beta \cdot A$

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<footer style = "font-size: 18px">Matrix lecture-2 $\rightarrow$ Scalar multiplication $\rightarrow$ <span style = "color:blue"> Property $(\alpha+\beta) \cdot A$ </span> $\rightarrow$ Property $\alpha \cdot(A+B)$ $\rightarrow$ Proof </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-1"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-property-alpha-cdotab-primary"><primary style="font-size:24px"> Property $\alpha \cdot(A+B)$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>For any scalar $\alpha$ and for any two matrices $A$ and $B$ of same order, $\alpha \cdot(A+B)=\alpha \cdot A+\alpha \cdot B$.</p>
</li>
<li>
<p>Proof:</p>
</li>
<li>
<p>Let, $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$.</p>
</li>
<li>
<p>$\alpha \cdot(A+B) =\alpha \cdot\left(\left[a_{i j}\right]+\left[b_{i j}\right]\right)$</p>
</li>
<li>
<p>$ =\alpha \cdot\left(\left[a_{i j}+b_{i j}\right]\right)$</p>
</li>
<li>
<p>$ =\left[\alpha \cdot\left(a_{i j}+b_{i j}\right)\right]$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Scalar multiplication  $\rightarrow$ Property $(\alpha+\beta) \cdot A$ $\rightarrow$ <span style = "color:blue"> Property $\alpha \cdot(A+B)$ </span> $\rightarrow$ Proof $\rightarrow$ Property </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-3"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-property-primary"><primary style="font-size:24px"> Property </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>$(\alpha \cdot \beta) \cdot A =\alpha \cdot(\beta \cdot A)=\beta \cdot(\alpha \cdot A)$</p>
</li>
<li>
<p>$(\alpha \cdot \beta) \cdot A  =(\alpha \cdot \beta) \cdot\left[a_{i j}\right]$</p>
</li>
<li>
<p>$ =\left[(\alpha \beta) \cdot a_{i j}\right]$</p>
</li>
<li>
<p>$ =\left[\alpha \cdot\left(\beta \cdot a_{i j}\right)\right]$</p>
</li>
<li>
<p>$ =\alpha\left[\beta \cdot a_{i j}\right]$</p>
</li>
<li>
<p>$ =\alpha \cdot\left(\left[\beta \cdot a_{i j}\right]\right)$</p>
</li>
<li>
<p>$ =\alpha \cdot\left(\beta \cdot\left[a_{i j}\right]\right)$</p>
</li>
<li>
<p>$ =\alpha \cdot(\beta \cdot A) .$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Property $\alpha \cdot(A+B)$  $\rightarrow$ Proof $\rightarrow$ <span style = "color:blue"> Property </span> $\rightarrow$ Transpose of a matrix $\rightarrow$ Transpose of a matrix example </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-5"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-transpose-of-a-matrix-example-primary"><primary style="font-size:24px"> Transpose of a matrix example </primary></h3>
<hr>
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<div style='float: left; width:99%; font-size:30px; text-align:left'>
<ul>
<li>
<p>If $A$ is a matrix of order $n \times m$, then $A^t$ is a matrix of order $m \times n$.</p>
</li>
<li>
<p>1)$\quad$ A = <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>7</mn> </msqrt> </mfrac> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></p>
</li>
<li>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msup> <mi>A</mi> <mi>t</mi> </msup> <mo>=</mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mtd> <mtd> <mfrac> <mn>1</mn> <msqrt> <mn>7</mn> </msqrt> </mfrac> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Property $\rightarrow$ Transpose of a matrix $\rightarrow$ <span style = "color:blue"> Transpose of a matrix example </span> $\rightarrow$ Example of transpose of a matrix $\rightarrow$ Property $ (A+B)^t=A^t+B^t$  </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-6"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-of-transpose-of-a-matrix-primary"><primary style="font-size:24px"> Example of transpose of a matrix </primary></h3>
<hr>
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<div style='float: left; width:99%; font-size:30px;text-align:left'>
<ul>
<li>
<ol start="2">
<li>A= <math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mo>+</mo> <mn>5</mn> <mi>i</mi> </mtd> <mtd> <mn>3</mn>  </mtd> <mtd>  <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></li>
</ol>
</li>
<li>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"> <mstyle displaystyle="true"> <msup> <mi>A</mi> <mi>t</mi> </msup> <mo>=</mo> <mrow> <mo>[</mo> <mtable columnlines="none none none"> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>4</mn> <mo>+</mo> <mn>5</mn> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>2</mn> <mi>i</mi> </mtd> <mtd> <mn>3</mn> </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>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mstyle> </math></p>
</li>
</ul>
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</main>
<hr>
<footer style = "font-size: 18px">Transpose of a matrix $\rightarrow$ Transpose of a matrix example $\rightarrow$ <span style = "color:blue"> Example of transpose of a matrix </span> $\rightarrow$ Property $ (A+B)^t=A^t+B^t$ $\rightarrow$  Property $ (\alpha A)^t=\alpha \cdot A^t$ </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-7"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-property--abtatbt-primary"><primary style="font-size:24px"> Property $ (A+B)^t=A^t+B^t$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:28px'>
<ul>
<li>
<ol>
<li>$ (A+B)^t=A^t+B^t. $</li>
</ol>
</li>
<li>
<p>Proof:</p>
</li>
<li>
<p>Let, $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$</p>
</li>
<li>
<p>$(A+B)^t=\left(\left[a_{i j}\right]+\left[b_{i j}\right]\right)^t=\left(\left[a_{i j}+b_{i j}\right]\right)^t$</p>
</li>
<li>
<p>$ =\left[a_{j i}+b_{j i}\right]$</p>
</li>
<li>
<p>$ =\left[a_{j i}\right]+\left[b_{j i}\right]$</p>
</li>
<li>
<p>$ =A^t+B^t .$</p>
</li>
<li>
<p>$\therefore(A+B)^t  =A^t+B^t .$</p>
</li>
</ul>
</div>
</main>
<hr>
<footer style = "font-size: 18px">Transpose of a matrix example $\rightarrow$ Example of transpose of a matrix $\rightarrow$ <span style = "color:blue"> Property $ (A+B)^t=A^t+B^t$ </span> $\rightarrow$ Property $ (\alpha A)^t=\alpha \cdot A^t$ $\rightarrow$ Symmetric and skew-symmetric matrix </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-8"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-property--alpha-atalpha-cdot-at-primary"><primary style="font-size:24px"> Property $ (\alpha A)^t=\alpha \cdot A^t$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<ol start="2">
<li>For any scalar $\alpha$,  any matrix $A,(\alpha A)^t=\alpha \cdot A^t$.</li>
</ol>
</li>
<li>
<p>Proof:</p>
</li>
<li>
<p>Let, $A=[a_{ij}]$</p>
</li>
<li>
<p>$(\alpha.A)^t = (\alpha .[a_{ij}])^t=([\alpha . a_{ij}])^t$</p>
</li>
<li>
<p>$=[\alpha . a_{ij}]=\alpha.[a_{ij}]$</p>
</li>
<li>
<p>$=\alpha. A^t$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Example of transpose of a matrix $\rightarrow$ Property $ (A+B)^t=A^t+B^t$ $\rightarrow$ <span style = "color:blue"> Property $ (\alpha A)^t=\alpha \cdot A^t$ </span> $\rightarrow$ Symmetric and skew-symmetric matrix $\rightarrow$ Examples </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-9"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-symmetric-and-skew-symmetric-matrix-primary"><primary style="font-size:24px"> Symmetric and skew-symmetric matrix </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>Definition : $A$ is called a symmetric matrix if $A=A^t$ and a matrix $A$ is called a skew-symmetric matrix if $A=-A^t$.</p>
</li>
<li>
<p>$A=\left[\begin{array}{lll}1 & 2 & 3 \\2 & 3 & 4 \\3 & 4 & 5\end{array}\right]$</p>
</li>
<li>
<p>$ A^t=\left[\begin{array}{lll}1 & 2 & 3 \\2 & 3 & 4 \\3 & 4 & 5\end{array}\right] $</p>
</li>
<li>
<p>$\quad \text { Note that } A=A^t.$</p>
</li>
<li>
<p>$\quad \therefore A \text { is symmetric. } $</p>
</li>
</ul>
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</main>
<hr>
<footer style = "font-size: 18px">Property $\alpha \cdot(A+B)$ $\rightarrow$ Transpose of a matrix example $\rightarrow$ <span style = "color:blue"> Property $ (\alpha A)^t=\alpha \cdot A^t$ </span> $\rightarrow$ Examples $\rightarrow$ $A+A^t$ symmetric matrix </footer>

### <Success style="font-size:25px"> Matrices L-2 </Success>
### <primary style="font-size:24px"> $A+A^t$ symmetric matrix </primary>
<hr>
<main>

- Proof:

- Let, $A=\left[a_{ij}\right]$. Then, $A^t=\left[a_{j i}\right]$.

- $ A+A^t  =\left[a_{i j}\right]+\left[a_{j i }\right]  =\left[a_{i j}+a_{j i}\right]  \longrightarrow(1) $

- $ \left(A+A^t\right)^t  =\left(\left[a_{i j}+a_{j i}\right]\right)^t  =\left[a_{j i}+a_{i j}\right] $

- $ =\left[a_{i j}+a_{j i}\right] \quad =A+A^t \quad \text { (From (1) } $

- Thus, $A+A^t$ is symmetric.

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<footer style = "font-size: 18px">Property $ (\alpha A)^t=\alpha \cdot A^t$ $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> $A+A^t$ symmetric matrix </span> $\rightarrow$ $A-A^t$ skew-symmetric matrix $\rightarrow$ $A-A^t$ skew-symmetric matrix </footer>

### <Success style="font-size:25px"> Matrices L-2 </Success>
### <primary style="font-size:24px"> $A-A^t$ skew-symmetric matrix </primary>
<hr>
<main>

- If $A$ is any square matrix, then $A-A^t$ is a skew-symmetric matrix.

- Proof:

- Let, $A=\left[a_{i j}\right]$. Then $A^t=\left[a_{j i}\right]$.

- $ A - A^t  =\left[a_{i j}\right]-\left[a_{j i}\right] $

- $ =\left[a_{i j}\right]+\left[-a_{j i}\right] =\left[a_{i j}-a_{j i}\right] \rightarrow(1) . $

- $ \left(A-A^t\right)^t  =\left(\left[a_{i j}-a_{j i}\right]\right)^t $

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<hr>
<footer style = "font-size: 18px">Examples $\rightarrow$ $A+A^t$ symmetric matrix $\rightarrow$ <span style = "color:blue"> $A-A^t$ skew-symmetric matrix </span> $\rightarrow$ $A-A^t$ skew-symmetric matrix $\rightarrow$ Theorem </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-10"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-a-at-skew-symmetric-matrix-primary"><primary style="font-size:24px"> $A-A^t$ skew-symmetric matrix </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>$ =\left[-a_{j i}+a_{i j}\right] $</p>
</li>
<li>
<p>$ =\left[-\left(a_{i j}-a_{j i}\right)\right] $</p>
</li>
<li>
<p>$ =-\left[a_{i j}-a_{j i}\right] $</p>
</li>
<li>
<p>$ =-\left(A-A^t\right) \quad(\text { From (1)). } $</p>
</li>
<li>
<p>Thus, $A-A^t$ is a skew-symmetric matrix.</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">$A+A^t$ symmetric matrix $\rightarrow$ $A-A^t$ skew-symmetric matrix $\rightarrow$ <span style = "color:blue"> $A-A^t$ skew-symmetric matrix </span> $\rightarrow$ Theorem $\rightarrow$ Theorem contd... </footer>

### <Success style="font-size:25px"> Matrices L-2 </Success>
### <primary style="font-size:24px"> Multiplication of matrices </primary>
<hr>
<main>

- Let, $A$ be a matrix of order $m \times n$ and $B$ be a matrix of order $n \times r$. Then the product
- of $A$ & $B$, denoted $A B$, is obtained as follows:

- $ A=\left[a_{i j}\right]_{\substack{ 1 \leqslant i \leqslant m \},{\ 1 \leqslant j \leqslant n }} $

- $ B=\left[b_{i j}\right]_{\substack{ 1 \leqslant i \leqslant n \},{\ 1 \leqslant j \leqslant r }} $

- Let, $A B=c=\left[c_{i j}\right]$. Then,

- $c_{i j}=\sum_{k=1}^n a_{i k} b_{k j} \text {. }$
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<footer style = "font-size: 18px">Theorem contd... $\rightarrow$ Theorem contd... $\rightarrow$ <span style = "color:blue"> Multiplication of matrices </span> $\rightarrow$ Example of multiplication of matrices $\rightarrow$ Example of multiplication of matrices </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-13"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-of-multiplication-of-matrices-primary"><primary style="font-size:24px"> Example of multiplication of matrices </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>$ A=\left[\begin{array}{ll}1 & 2 \\3 & 4\end{array}\right] $</p>
</li>
<li>
<p>$ B=\left[\begin{array}{l}5 \\ 6\end{array}\right]$</p>
</li>
<li>
<p>$ m=2 $, $ n=2 $, $ r=1 $</p>
</li>
<li>
<p>order of A = 2 x 2</p>
</li>
<li>
<p>order of B = 2 x 1</p>
</li>
</ul>
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<div style='float: right; width:01%;'>

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<hr>
<footer style = "font-size: 18px">Theorem contd... $\rightarrow$ Multiplication of matrices $\rightarrow$ <span style = "color:blue"> Example of multiplication of matrices </span> $\rightarrow$ Example of multiplication of matrices $\rightarrow$ Example $A_ {\tiny 2 × 2} \cdot B_{\tiny 2 × 3}$ </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-14"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-of-multiplication-of-matrices-primary-1"><primary style="font-size:24px"> Example of multiplication of matrices </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>$ \therefore\left[\begin{array}{ll}1 & 2 \\3 & 4\end{array}\right] \left[\begin{array}{l}5 \\ 6\end{array}\right]$</p>
</li>
<li>
<p>$ =\left[\begin{array}{ll}\sum_1^2 a_{1k} b_{k1} \\ \sum_1^2 a_{2k} b_{k-1}\end{array}\right] $</p>
</li>
<li>
<p>$ =\left[\begin{array}{ll}a_{11} b_{11} + a_{12} b_{21} \\a_{21} b_{11} + a_{22} b_{21}\end{array}\right] $</p>
</li>
<li>
<p>$ =\left[\begin{array}{c}5+12 \\15+24\end{array}\right] $</p>
</li>
<li>
<p>$ =\left[\begin{array}{l}17 \\39\end{array}\right] $</p>
</li>
</ul>
</div>
</main>
<hr>
<footer style = "font-size: 18px">Multiplication of matrices $\rightarrow$ Example of multiplication of matrices $\rightarrow$ <span style = "color:blue"> Example of multiplication of matrices </span> $\rightarrow$ Example $A_ {\tiny 2 × 2} \cdot B_{\tiny 2 × 3}$ $\rightarrow$ Example $A_ {\tiny 3 × 3} \cdot B_{\tiny 3 × 2}$ </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-15"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-a_-tiny-2--2-cdot-b_tiny-2--3-primary"><primary style="font-size:24px"> Example $A_ {\tiny 2 × 2} \cdot B_{\tiny 2 × 3}$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px; text-align:left'>
<ul>
<li>
<p>$A =\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] _ {\tiny 2 × 2} \\$</p>
</li>
<li>
<p>$B=\left[\begin{array}{ccc}1 & 3 & 5 \\ 2 & 4 & 6\end{array}\right]_{\tiny 2 × 3} $</p>
</li>
<li>
<p>$A B  =\left[\begin{array}{lll}1+4 & 3+8 & 5+12 \\ 3+8 & 9+16 & 15+24\end{array}\right] $</p>
</li>
<li>
<p>$ =\left[\begin{array}{lll}5 & 11 & 17 \\ 11 & 25 & 39\end{array}\right] $</p>
</li>
</ul>
</div>
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<hr>
<footer style = "font-size: 18px">Example of multiplication of matrices $\rightarrow$ Example of multiplication of matrices $\rightarrow$ <span style = "color:blue"> Example $A_ {\tiny 2 × 2} \cdot B_{\tiny 2 × 3}$ </span> $\rightarrow$ Example $A_ {\tiny 3 × 3} \cdot B_{\tiny 3 × 2}$ $\rightarrow$ $ (A B)^t=B^t A^t $ </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-16"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-a_-tiny-3--3-cdot-b_tiny-3--2-primary"><primary style="font-size:24px"> Example $A_ {\tiny 3 × 3} \cdot B_{\tiny 3 × 2}$ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>$ A=\left[\begin{array}{lll}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{array}\right]_{3×3} $</p>
</li>
<li>
<p>$B=\left[\begin{array}{ll}1 & 2 \\3 & 4 \\5 & 6\end{array}\right]_{3×2} $</p>
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<p>$AB =\left[\begin{array}{lll}1+6+15 & 2+8+18 \\4+15+30 & 8+20+36 \\7+24+45 & 14+32+54\end{array}\right]  $$ =\left[\begin{array}{lll}22 & 28 \\49 & 64 \\76 & 100\end{array}\right] $</p>
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<hr>
<footer style = "font-size: 18px">Example of multiplication of matrices $\rightarrow$ Example $A_ {\tiny 2 × 2} \cdot B_{\tiny 2 × 3}$ $\rightarrow$ <span style = "color:blue"> Example $A_ {\tiny 3 × 3} \cdot B_{\tiny 3 × 2}$ </span> $\rightarrow$ $ (A B)^t=B^t A^t $ $\rightarrow$ $ (A B)^t=B^t A^t $ </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-17"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px--a-btbt-at--primary"><primary style="font-size:24px"> $ (A B)^t=B^t A^t $ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>For any two square matrices $A$ & $B$ of same order,</p>
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<p>$ (A B)^t=B^t A^t \text {. } $</p>
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<p>Proof:</p>
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<p>Let,$ A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right], 1 \leqslant i, j \leqslant n $.</p>
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<p>$ \text { Let } C=A B=\left[c_{i j}\right] $</p>
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<p>where, $ c_{i j}=\sum_{k=1}^n a_{i k} b_{k j} $.</p>
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<p>$ (A B)^t=c^t=\left[c_{i j}\right]^t=\left[c_{j i}\right]=\left[\sum_{k=1}^n a_{j k} b_{k i}\right] $ .</p>
</li>
</ul>
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example $A_ {\tiny 2 × 2} \cdot B_{\tiny 2 × 3}$ $\rightarrow$ Example $A_ {\tiny 3 × 3} \cdot B_{\tiny 3 × 2}$ $\rightarrow$ <span style = "color:blue"> $ (A B)^t=B^t A^t $ </span> $\rightarrow$ $ (A B)^t=B^t A^t $ $\rightarrow$ Example to previous property </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-18"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px--a-btbt-at--primary-1"><primary style="font-size:24px"> $ (A B)^t=B^t A^t $ </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
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<p>$ B^t A^t  =\left[b_{i j}\right]^t\left[a_{i j}\right]^t $</p>
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<p>$ =\left[b_{j i}\right]\left[a_{j i}\right] $</p>
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<p>$  =\left[\sum_{k=1}^n b_{k i} a_{k j i}\right] $</p>
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<p>$ =\left[\sum_{n=1}^n a_{k j} b_{k i}\right] $</p>
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<p>$ =(A B)^t .$</p>
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<p>Thus, $ (A B)^t  =B^t A^t .$</p>
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</ul>
</div>
<div style='float: right; width:01%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Example $A_ {\tiny 3 × 3} \cdot B_{\tiny 3 × 2}$ $\rightarrow$ $ (A B)^t=B^t A^t $ $\rightarrow$ <span style = "color:blue"> $ (A B)^t=B^t A^t $ </span> $\rightarrow$ Example to previous property $\rightarrow$ Example to previous property contd... </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-19"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-to-previous-property-primary"><primary style="font-size:24px"> Example to previous property </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px;text-align:left; '>
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<li>
<p>$ A =\left[\begin{array}{lll}1 & 2 & 3 \\4 & 5 & 6\end{array}\right]_{2×3} $</p>
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<p>$B=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\5 & 6 & 7 & 8 \\1 & 2 & 4 & 5\end{array}\right]_{3×4} $</p>
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<li>
<p>$ A B=\left[\begin{array}{llll}1+10+3 & 2+12+6 & 3+14+12 & 4+16+15 \\4+25+6 & 8+30+12 & 12+35+24 & 16+40+30\end{array}\right] $</p>
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<p>$ =\left[\begin{array}{llll}14 & 20 & 29 & 35 \\35 & 50 & 11 & 86\end{array}\right]_ {2\times4 }$</p>
</li>
</ul>
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<div style='float: right; width:1%;'>

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<hr>
<footer style = "font-size: 18px">$ (A B)^t=B^t A^t $ $\rightarrow$ $ (A B)^t=B^t A^t $ $\rightarrow$ <span style = "color:blue"> Example to previous property </span> $\rightarrow$ Example to previous property contd... $\rightarrow$ Example to previous property contd... </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-20"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-to-previous-property-contd-primary"><primary style="font-size:24px"> Example to previous property contd… </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:28px;text-align:left'>
<ul>
<li>
<p>$(A B)^t=\left[\begin{array}{ll}14 & 35 \\20 & 50 \\29 & 71 \\38 & 86\end{array}\right] $</p>
</li>
<li>
<p>$B^t=\left[\begin{array}{lll}1 & 5 & 1 \\ 2 & 6 & 2 \\ 3 & 7 & 4 \\ 4 & 8 & 5\end{array}\right] $</p>
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<li>
<p>$A^t=\left[\begin{array}{ll}1 & 4 \\ 2 & 5 \\ 3 & 6\end{array}\right] $</p>
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</ul>
</div>
</main>
<hr>
<footer style = "font-size: 18px">$ (A B)^t=B^t A^t $ $\rightarrow$ Example to previous property $\rightarrow$ <span style = "color:blue"> Example to previous property contd... </span> $\rightarrow$ Example to previous property contd... $\rightarrow$ Example to previous property contd... </footer>

<h3 id="success-stylefont-size25px-matrices-l-2-success-21"><Success style="font-size:25px"> Matrices L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-example-to-previous-property-contd-primary-1"><primary style="font-size:24px"> Example to previous property contd… </primary></h3>
<hr>
<main>
<div style='float: left; width:99%; font-size:30px'>
<ul>
<li>
<p>$B^t A^t=\left[\begin{array}{lll}1 & 5 & 1 \\ 2 & 6 & 2 \\ 3 & 7 & 4 \\ 4 & 8 & 5\end{array}\right]\left[\begin{array}{ll}1 & 4 \\ 2 & 5 \\ 3 & 6\end{array}\right] $</p>
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<p>$ =\left[\begin{array}{lll}1+10+3 & 4+25+6 \\ 2+12+6 & 8+30+12 \\ 3+14+12 & 12+35+24 \\ 4+16+15 & 16+40+30\end{array}\right] $</p>
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<hr>
<footer style = "font-size: 18px">Example to previous property $\rightarrow$ Example to previous property contd... $\rightarrow$ <span style = "color:blue"> Example to previous property contd... </span> $\rightarrow$ Example to previous property contd... $\rightarrow$ Thank you </footer>