mathematics-class-12-unit-04-chapter-02-determinants-l-2-11-i4vozlyadvc

### <Success style="font-size:25px"> Determinants L-2 </Success>
### <primary style="font-size:24px"> Properties of determinants-1  </primary>
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- Property 1. If the entire row (or column ) of a square matrix  is zero, then the value of its determinant is also zero

- Example:

- $\left[\begin{array}{ll}0 & 0 \\\c & d\end{array}\right] $

- $ \rightarrow \frac{0 \times()}{0} + 0 \times( ) $
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<footer style = "font-size: 18px">Determinants lecture -2 $\rightarrow$ Recap $\rightarrow$ <span style = "color:blue"> Properties of determinants-1  </span> $\rightarrow$ Properties of determinants-2 $\rightarrow$ Properties of determinants-3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-2-success-1"><Success style="font-size:25px"> Determinants L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-properties-of-determinants-2-primary"><primary style="font-size:24px"> Properties of determinants-2 </primary></h3>
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<p>Property 2: For a diagonal matrix, the determinant is the product of the diagonal entries</p>
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<p>Example:</p>
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<p>$ \left|\left[\begin{array}{lll}a_{11} & 0 & 0 \\0 & a_{22} & 0 \\0 & 0 & a_{33}\end{array}\right]\right| $</p>
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<p>$ =a_{11}\left|\begin{array}{cc}a_{22} & 0 \\0 & a_{33}\end{array}\right|+0+0 $</p>
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<p>$ =\underline{a_{11}  a_{12}  a_{13} } $</p>
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<footer style = "font-size: 18px">Recap $\rightarrow$ Properties of determinants-1 $\rightarrow$ <span style = "color:blue"> Properties of determinants-2  </span> $\rightarrow$ Properties of determinants-3 $\rightarrow$ Properties of determinants-3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-2-success-2"><Success style="font-size:25px"> Determinants L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-properties-of-determinants-3-primary"><primary style="font-size:24px"> Properties of determinants-3 </primary></h3>
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<p>Property 3: For a triangular matrix, the determinant is the product of diagonal entries</p>
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<p>Example:</p>
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<p>a) $\left|\left[\begin{array}{cc}-a & b \\ 0 & c\end{array}\right]\right|=a c $, just the product of the diagonal entries</p>
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<p>b) $\left|\left[\begin{array}{lll}a & 0 & 0 \\ \cdots & b & 0 \\ \cdots & \cdots & c\end{array}\right]\right| $</p>
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<p>$ =a\left|\begin{array}{ll}b & 0 \\ \cdot & c\end{array}\right| $</p>
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<p>$ =a b c $, again the product of diagonal</p>
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<footer style = "font-size: 18px">Properties of determinants-1 $\rightarrow$ Properties of determinants-2  $\rightarrow$ <span style = "color:blue"> Properties of determinants-3 </span> $\rightarrow$ Properties of determinants-3 $\rightarrow$ Properties of determinants-4 </footer>

<h3 id="success-stylefont-size25px-determinants-l-2-success-3"><Success style="font-size:25px"> Determinants L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-properties-of-determinants-3-primary-1"><primary style="font-size:24px"> Properties of determinants-3 </primary></h3>
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<p>Note :</p>
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<p>$\left[\left(\begin{array}{cc}0 & 0 \\ a & b\end{array}\right)\right]$</p>
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<p>$\therefore$ area enclosed by the column vectors is 0</p>
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<p>Consistent with the determinant being 0</p>
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<footer style = "font-size: 18px">Properties of determinants-2  $\rightarrow$ Properties of determinants-3 $\rightarrow$ <span style = "color:blue"> Properties of determinants-3 </span> $\rightarrow$ Properties of determinants-4 $\rightarrow$ Properties of determinants-5 </footer>

<h3 id="success-stylefont-size25px-determinants-l-2-success-4"><Success style="font-size:25px"> Determinants L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-properties-of-determinants-4-primary"><primary style="font-size:24px"> Properties of determinants-4 </primary></h3>
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<p>Property 4. Interchanging the row and column of a square matrix, IT does not change the value of determinant</p>
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<p>$\operatorname{det}(A)=\operatorname{det}\left(A^{\top}\right)$</p>
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<p>Example</p>
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<p>$  {\left[\begin{array}{ll}a & b \\c & d\end{array}\right] \text { transpose }\left[\begin{array}{ll}a & c \\b & d\end{array}\right]} $</p>
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<p>$  a d-b c \stackrel{\text { same }}{\longleftrightarrow} a d-b c $</p>
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<footer style = "font-size: 18px">Properties of determinants-3 $\rightarrow$ Properties of determinants-3 $\rightarrow$ <span style = "color:blue"> Properties of determinants-4 </span> $\rightarrow$ Properties of determinants-5 $\rightarrow$ Properties of determinants-6 </footer>

### <Success style="font-size:25px"> Determinants L-2 </Success>
### <primary style="font-size:24px"> Properties of determinants-5 </primary>
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- Property 5: If any two rows (or columns) are interchanged, then the sign of the determinant changes

- Example:

- ${\left.\begin{array}{ll}
 R_1 \rightarrow \\\ 
 R_2 \rightarrow
\end{array}\right.} {\left[\begin{array}{ll}
 a & b \\\ 
 c & d
\end{array}\right]}$

- ${R_ 1 \leftrightarrow R_2 \Rightarrow\left[\begin{array}{ll}
 c & d \\\ 
 a & b
\end{array}\right]}$

- $\begin{aligned} & a d-b c , \quad c b-a d \\ & =-(a d-b c) \\ & \end{aligned}$
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<footer style = "font-size: 18px">Properties of determinants-3 $\rightarrow$ Properties of determinants-4 $\rightarrow$ <span style = "color:blue"> Properties of determinants-5 </span> $\rightarrow$ Properties of determinants-6 $\rightarrow$ Properties of determinants-6 </footer>

### <Success style="font-size:25px"> Determinants L-2 </Success>
### <primary style="font-size:24px"> Properties of determinants-6 </primary>
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- Property 6. If two rows (or columns) of a matrix are identical, then the value of its determinant is zero

- Proof: Consider square matrix $A$. with two identical rows $ R_i  R_j $

- $R_i \leftrightarrow R_j$ will give same matrix $A$

- But, from previous property, sign of determinant changes

- $-\operatorname{det}(A)=\operatorname{det}(\underbrace{A^{R_i \rightarrow R_j}}_A)$
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<footer style = "font-size: 18px">Properties of determinants-4 $\rightarrow$ Properties of determinants-5 $\rightarrow$ <span style = "color:blue"> Properties of determinants-6 </span> $\rightarrow$ Properties of determinants-6 $\rightarrow$ Properties of determinants-6 </footer>

<h3 id="success-stylefont-size25px-determinants-l-2-success-5"><Success style="font-size:25px"> Determinants L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-properties-of-determinants-6-primary"><primary style="font-size:24px"> Properties of determinants-6 </primary></h3>
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<p>Note  : Consider $3 \times 3$ matrix</p>
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<p>$\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right] $</p>
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<p>$A_{i j}$ is cofactor of $a_{i j}$</p>
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<p>$ d e t=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} $</p>
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<p>$ a_{11} A_{21}+a_{12} A_{22}+a_{13} A_{23}=? $</p>
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<p>$ a_{11} A_{21}+a_{12} A_{22}+a_{13} A_{23}=0$</p>
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<p>$ =a_{11}(-1)^{2+1}\left|\begin{array}{ll}a_{12} & a_{13} \\ a_{32} & a_{33}\end{array}\right|+a_{12}(-1)^{2+2}\left|\begin{array}{ll}a_{11} & a_{13} \\ a_{31} & a_{33}\end{array}\right| $ $+ a_{13}(-1)^{2+3}\left|\begin{array}{ll}a_{11} & a_{12} \\ a_{31} & a_{32}\end{array}\right| $</p>
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<footer style = "font-size: 18px">Properties of determinants-5 $\rightarrow$ Properties of determinants-6 $\rightarrow$ <span style = "color:blue"> Properties of determinants-6 </span> $\rightarrow$ Properties of determinants-6 $\rightarrow$ Properties of determinants-7 </footer>

<h3 id="success-stylefont-size25px-determinants-l-2-success-6"><Success style="font-size:25px"> Determinants L-2 </Success></h3>
<h3 id="primary-stylefont-size24px-properties-of-determinants-6-primary-1"><primary style="font-size:24px"> Properties of determinants-6 </primary></h3>
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<li>$ = -1[a_{11}\left|\begin{array}{ll}a_{12} & a_{13} \\ a_{32} & a_{33}\end{array}\right|] - $ $ [a_{12}\left|\begin{array}{ll}a_{11} & a_{13} \\ a_{31} & a_ {33}\end{array}\right|] +  [a_ {13}\left|\begin{array}{ll}a_ {11} & a_ {12} \\ a_ {31} & a_ {32}\end{array}\right|] =-\left|\left[\begin{array}{lll}a_{11} & a_ {12} & a_ {13} \\ a_ {11} & a_ {12} & a_ {13} \\ a_ {31} & a_ {32} & a_{33}\end{array}\right]\right|=0$</li>
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<footer style = "font-size: 18px">Properties of determinants-6 $\rightarrow$ Properties of determinants-6 $\rightarrow$ <span style = "color:blue"> Properties of determinants-6 </span> $\rightarrow$ Properties of determinants-7 $\rightarrow$ Properties of determinants-8 </footer>

### <Success style="font-size:25px"> Determinants L-2 </Success>
### <primary style="font-size:24px"> Properties of determinants-7 </primary>
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- Property 7: If each element of a row (or a column) is multiplied by a constant ' $k$ ' 'then the value of the determinant gets multiplied by '$k$'

- Example:

- $ {\left[\begin{array}{ll}a & b \\\c & d\end{array}\right] \xrightarrow{R_1 \rightarrow k R_1}\left[\begin{array}{rr}k a & k b\\\c & d\end{array}\right]} $

- $ \quad \downarrow $  $\qquad \qquad \qquad \qquad \downarrow $

- $ ad -bc \quad \quad \quad   k a(d)-k b(c) =k\left[a(d)-b (c)\right] $

- $\Rightarrow $ det scales by 'k'.
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<footer style = "font-size: 18px">Properties of determinants-6 $\rightarrow$ Properties of determinants-6 $\rightarrow$ <span style = "color:blue"> Properties of determinants-7 </span> $\rightarrow$ Properties of determinants-8 $\rightarrow$ Properties of determinants-9 </footer>

### <Success style="font-size:25px"> Determinants L-2 </Success>
### <primary style="font-size:24px"> Properties of determinants-8 </primary>
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- Property 8: If some or all elements of a new (or column)can be written as a sum of two terms, then the determinant can be expressed as the sum of determinants of matrices obtained by separating the origin matrix

- Example:

- $ \left|\left[\begin{array}{cc}a+x & b+y \\\ c & d\end{array}\right]\right|=\left|\begin{array}{ll}a & b \\\c & d\end{array}\right|+\left|\begin{array}{ll}x & y \\\c & d\end{array}\right| $

- $ \(a+x) (d)+\(b+y) (-c) $

- $ =\{a(d)+b (-c)} +\{x(d)+y (-c)\} $

- $ = {det}\left[\begin{array}{ll}a & b \\\c & d\end{array}\right] + {det}\left[\begin{array}{ll}x & y \\\c & d\end{array}\right]$

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<footer style = "font-size: 18px">Properties of determinants-6 $\rightarrow$ Properties of determinants-7 $\rightarrow$ <span style = "color:blue"> Properties of determinants-8 </span> $\rightarrow$ Properties of determinants-9 $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Determinants L-2 </Success>
### <primary style="font-size:24px"> Properties of determinants-9 </primary>
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- Property 9: If each element of a row(or column) is replaced by a sum of that element and an element of another row or column), then the value of the determinant remains the some

- Example:

- $\left[\begin{array}{ll}a & b \\\c & d\end{array}\right] \xrightarrow{R_ 1 \rightarrow R_1+R_2}  \left[\begin{array}{cc}a+c & b+d \\\c & d\end{array}\right] $

$\quad \quad \downarrow$ $\qquad \qquad \qquad \qquad \qquad \downarrow$

- $ a d-b c $ $\qquad \qquad \qquad $ $\left|\begin{array}{ll}a & b \\\c & d\end{array}\right|+ \left|\begin{array}{cc}c & d \\\c & d\end{array}\right| = \left|\begin{array}{ll}a & b \\\c & d\end{array}\right| + 0 $

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<footer style = "font-size: 18px">Properties of determinants-7 $\rightarrow$ Properties of determinants-8 $\rightarrow$ <span style = "color:blue"> Properties of determinants-9 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>