Evaluation of the Determinant:

PYQ-2023-Determinants-Q5

$\quad$ A square matrix of order 3, the value of the determinant is

$$(a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32})- (a_{13} a_{22} a_{31} + a_{11} a_{23} a_{32} + a_{12} a_{21} a_{33})$$

Minor:

$\quad$ Minor of an element is defined as the determinant obtained by deleting the row and column in which that element lies.

Cofactor:

$\quad$ Cofactor of an element $a_{i j}$ is related to its minor as $C_{i j}(-1)^{ i+ j}M_{ij}$ where ‘i’ denotes the i-th row and ‘j’ denotes the j-th column to which the element $a_{ij}$ belongs.

Row and Column Operations:

• $\quad R_i ↔ R_j$ or $C_i ↔ C_j$ when i ≠ j; This notation is used when we interchange i-th row (or column) and j-th row (or column).

• $\quad R_i ↔ C_j$ ; This converts the row into the corresponding column.

• $\quad R_i → k R_i$ or $C_i → kC_i$ ; k ∈ Real numbers; This represents multiplication of i-th row (or column) by k.

• $\quad R_i → k R_i + R_j$; (i ≠ j); This symbol is used to multiply i-th row (or column) by k and adding the j-th row (or column) to it.

PROPERTIES OF DETERMINANTS:

PYQ-2023-Determinants-Q1, PYQ-2023-Matrices-Q3, PYQ-2023-Matrices-Q8, PYQ-2023-Matrices-Q10, PYQ-2023-Matrices-Q14, PYQ-2023-AOD-Q9

• Reflection Property

  The determinant remains unaltered if its rows are changed into columns and the columns into rows.

• All-zero Property

  If all the elements of a row (or column) are zero, then the determinant is zero.

• Proportionality (Repetition) Property

  If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.

• Switching Property

  The interchange of any two rows (or columns) of the determinant changes its sign.

• Scalar Multiple Property

  If all the elements of a row (or column) of a determinant are multiplied by a nonzero constant, then the determinant gets multiplied by the same constant.

• Sum Property $$\left|\begin{array}{lll}a_1+b_1 & c_1 & d_1 \\ a_2+b_2 & c_2 & d_2 \\ a_3+b_3 & c_3 & d_3\end{array}\right|=\left|\begin{array}{lll}a_1 & c_1 & d_1 \\ a_2 & c_2 & d_2 \\ a_3 & c_3 & d_3\end{array}\right|+\left|\begin{array}{lll}b_1 & c_1 & d_1 \\ b_2 & c_2 & d_2 \\ b_3 & c_3 & d_3\end{array}\right|$$

• Property of Invariance $$\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=\left|\begin{array}{lll}a_1+\alpha b_1+\beta c_1 & b_1 & c_1 \\ a_2+\alpha b_2+\beta c_2 & b_2 & c_2 \\ a_3+\alpha b_3+\beta c_3 & b_3 & c_3\end{array}\right|$$

$\quad \quad \quad \quad $ That is, a determinant remains unaltered under an operation of the form $C_i \rightarrow C_i+\alpha C_j+\beta C_k$, where $j, k \neq i$, or

$\quad \quad \quad \quad $ An operation of the form $R_i \rightarrow R_i+\alpha R_j+\beta R_k$, where $j, k \neq i$

• Factor Property

  If a determinant $\Delta$ becomes zero when we put $\mathrm{x}=\alpha$, then $(\mathrm{x}-\alpha)$ is a factor of $\Delta$.

• Triangle Property

  If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements. That is, $$ \left|\begin{array}{ccc} a_1 & a_2 & a_3 \\ 0 & b_2 & b_3 \\ 0 & 0 & c_3 \end{array}\right|=\left|\begin{array}{ccc} a_1 & 0 & 0 \\ a_2 & b_2 & 0 \\ a_3 & b_3 & c_3 \end{array}\right|=a_1 b_2 c_3 $$

• Determinant of product of 2 matrices

$$ det(AB) = det(A) det(B)$$

Some Important Results:

• Three lines $a_1 x+b_1 y+c_1=0, a_2 x+b_2 y+c_2=0, a_3 x+b_3 y+c_3=0 $ are concurrent if

$$ \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0 $$

• $ a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ represents a pair of straight lines if $ a b c+2 f g h-a f^2-b g^2-c h^2=0=\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right| $

• Area of a triangle whose vertices are $\left(x_r, y_r\right) ; r=1,2,3$ is $ D=\frac{1}{2}\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{array}\right| $. If $D=0$ then the three points are collinear.

• Equation of a straight line passing through $ (x_1,y_1) \ \text{and} \ (x_2,y_2) $ is $\left|\begin{array}{lll}\mathrm{x} & \mathrm{y} & 1 \\ \mathrm{x}_1 & \mathrm{y}_1 & 1 \\ \mathrm{x}_2 & \mathrm{y}_2 & 1\end{array}\right|=0$.

• If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of the determinants.

  $$\left|\begin{array}{ccc}a_1+x & b_1+y & c_1+z \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|+\left|\begin{array}{ccc}x & y & z \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|$$

  It should be noted that while applying operations on determinants at least one row (or column) must remain unchanged i.e.

  Maximum number of simultaneous operations $=$ order of determinant -1

Differentiation Of Determinants:

• Let $\Delta(x)=\left|\begin{array}{ll}f_1(x) & g_1(x) \\ f_2(x) & g_2(x)\end{array}\right|$, where $f_1(x), f_2(x), g_1(x)$ and $g_2(x)$ are functions of $x$. Then,

$$ \Delta^{\prime}(\mathrm{x})=\left|\begin{array}{cc} \mathrm{f}_1^{\prime}(\mathrm{x}) & \mathrm{g}_1^{\prime}(\mathrm{x}) \\ \mathrm{f}_2(\mathrm{x}) & \mathrm{g}_2(\mathrm{x}) \end{array}\right|+\left|\begin{array}{cc} \mathrm{f}_1(\mathrm{x}) & \mathrm{g}_1(\mathrm{x}) \\ \mathrm{f}_2^{\prime}(\mathrm{x}) & \mathrm{g}_2^{\prime}(\mathrm{x}) \end{array}\right| \text { Also, } \Delta^{\prime}(\mathrm{x})=\left|\begin{array}{cc} \mathrm{f}_1^{\prime}(\mathrm{x}) & \mathrm{g}_1(\mathrm{x}) \\ \mathrm{f}_2^{\prime}(\mathrm{x}) & \mathrm{g}_2(\mathrm{x}) \end{array}\right|+\left|\begin{array}{cc} \mathrm{f}_1(\mathrm{x}) & \mathrm{g}_1^{\prime}(\mathrm{x}) \\ \mathrm{f}_2(\mathrm{x}) & \mathrm{g}_2^{\prime}(\mathrm{x}) \end{array}\right| $$

• If we write $\Delta(x)=\left[\begin{array}{ll}C_1 & C_2\end{array}\right]$, where $C_i$ denotes the $i^{\text {th }}$ column, then $$\Delta^{\prime}(x)=\left[\begin{array}{ll}C_1^{\prime} & C_2\end{array}\right]+\left[\begin{array}{ll}C_1 & C_2^{\prime}\end{array}\right]$$ $\quad \quad \quad $ where $C_i^{\prime}$ denotes the column obtained by differentiating functions in the $i^{\text {th }}$ column $C_i$.

• if $\Delta(x)=\left[\begin{array}{l}R_1 \\ R_2\end{array}\right]$, then $\Delta^{\prime}(x)=\left[\begin{array}{l}R_1{ }^{\prime} \\ R_2\end{array}\right]+\left[\begin{array}{l}R_1 \\ R_2^{\prime}\end{array}\right]$ Similarly, we can differentiate determinants of higher order.

Integration Of Determinants:

$\quad$ If $f(x), g(x)$ and $h(x)$ are functions of $x$ and $a, b, c, \alpha, \beta$ and $\gamma$ are constants such that

$$ \Delta(x)=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ a & b & c \\ \alpha & \beta & \gamma\end{array}\right| $$

$\quad$ then the integration of $\Delta(x)$ is given by

$$ \int \Delta(x) d x=\left|\begin{array}{ccc}\int f(x) d x & \int g(x) d x & \int h(x) d x \\ a & b & c \\ \alpha & \beta & \gamma\end{array}\right| $$

System of equations with 3 variables:

PYQ-2023-Determinants-Q1, PYQ-2023-Determinants-Q2, PYQ-2023-Determinants-Q3, PYQ-2023-Determinants-Q4, PYQ-2023-Determinants-Q5, PYQ-2023-Determinants-Q6, PYQ-2023-Determinants-Q8, PYQ-2023-Probability-Q1

$$ a_1 x+b_1 y+c_1 z=d_1\\ a_2 x+b_2 y+c_2 z=d_2\\ a_3 x+b_3 y+c_3 z=d_3 $$

$\quad$ To solve this system we first define the following determinants

$$ \Delta=\left|\begin{array}{lll} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right|, \Delta_1=\left|\begin{array}{lll} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{array}\right|, \Delta_2=\left|\begin{array}{lll} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \end{array}\right|, \Delta_3=\left|\begin{array}{lll} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \end{array}\right| $$

$\quad$ Now, to solve the system (Criterion For Consistency)

$\quad$ Check value of $\Delta$

• If $\Delta \neq 0$ then, Consistent system and has unique solution PYQ-2023-Determinants-Q2 PYQ-2023-Determinants-Q4

$$ \mathrm{x}=\frac{\Delta_1}{\Delta} ; \mathrm{y}=\frac{\Delta_2}{\Delta} ; \mathrm{z}=\frac{\Delta_3}{\Delta} $$

• $\Delta = 0$ then, check values of $\Delta_1, \Delta_2$ and $\Delta_3$

• Atleast one of $\Delta_1, \Delta_2$ and $\Delta_3$ is not zero then, the system is inconsistent. PYQ-2023-Determinants-Q3 PYQ-2023-Determinants-Q7

• If all $\Delta_1, \Delta_2$ and $\Delta_3$ are zero then, put $z=t$ and solve any two equations to get the values of $x$ and $y$ in terms of $t$