JEE Main & Advanced

Let us consider three homogeneous linear equations   \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z=0\],\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z=0\]   and  \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z=0\]   Eliminated \[x,\,\,y,\,\,z\] from above three equations we obtain   \[{{a}_{1}}({{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}})-{{b}_{1}}({{a}_{2}}{{c}_{3}}-{{a}_{3}}{{c}_{2}})+{{c}_{1}}({{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}})=0\]   …..(i)   The L.H.S. of (i) is represented by  \[\left| \,\begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|={{a}_{1}}\,\left| \,\begin{matrix}{{b}_{2}} & {{c}_{2}}  \\{{b}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|-{{b}_{1}}\,\left| \,\begin{matrix}{{a}_{2}} & {{c}_{2}}  \\{{a}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|+{{c}_{1}}\,\left| \,\begin{matrix}{{a}_{2}} & {{b}_{2}}  \\{{a}_{3}} & {{b}_{3}}  \\\end{matrix}\, \right|\]   Its contains three rows and three columns, it is called a determinant of third order.   The number of elements in a second order is \[{{2}^{2}}=4\] and the number of elements in a third order determinant is \[{{3}^{2}}=9\].   Rows and columns of a determinant : In a determinant horizontal lines counting from top \[{{1}^{st}},\text{ }{{2}^{nd}},\text{ }{{3}^{rd}},\ldots ..\] respectively known as rows and denoted by \[{{R}_{1}},\,\,{{R}_{2}},\,\,{{R}_{3}},\,\,......\] and vertical lines counting left to right, \[{{1}^{st}},\text{ }{{2}^{nd}},\text{ more...

P-1 : The value of determinant remains unchanged, if the rows and the columns are interchanged.   Since the determinant remains unchanged when rows and columns are interchanged, it is obvious that any theorem which is true for ‘rows’ must also be true for ‘columns’.   P-2 : If any two rows (or columns) of a determinant be interchanged, the determinant is unaltered in numerical value but is changed in sign only.   P-3 : If a determinant has two rows (or columns) identical, then its value is zero.   P-4 : If all the elements of any row (or column) be multiplied by the same number, then the value of determinant is multiplied by that number.   P-5 : 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.   more...

  (1) Minor of an element : If we take the element of the determinant and delete (remove) the row and column containing that element, the determinant left is called the minor of that element. It is denoted by \[{{M}_{ij}}\].   Consider the determinant \[\Delta =\left| \,\begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}}  \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\],   then determinant of minors \[M=\left| \,\begin{matrix} {{M}_{11}} & {{M}_{12}} & {{M}_{13}}  \\ {{M}_{21}} & {{M}_{22}} & {{M}_{23}}  \\ {{M}_{31}} & {{M}_{32}} & {{M}_{33}}  \\ \end{matrix}\, \right|\]   where  \[{{M}_{11}}=\] minor of \[{{a}_{11}}=\left| \,\begin{matrix} {{a}_{22}} & {{a}_{23}}  \\ {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\]  \[{{M}_{12}}=\]minor of  \[{{a}_{12}}=\left| \,\begin{matrix} {{a}_{21}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{33}}  \\ \end{matrix}\, \right|\] \[{{M}_{13}}=\] minor of \[{{a}_{13}}=\left| \,\begin{matrix} {{a}_{21}} & {{a}_{22}}  \\  {{a}_{31}} & {{a}_{32}}  \\ \end{matrix}\, \right|\]   Similarly, we can find the minors of other elements more...

 Let the two determinants of third order be,   \[{{D}_{1}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\] and \[{{D}_{2}}=\left| \,\begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} & {{\gamma }_{1}}  \\ {{\alpha }_{2}} & {{\beta }_{2}} & {{\gamma }_{2}}  \\ {{\alpha }_{3}} & {{\beta }_{3}} & {{\gamma }_{3}}  \\ \end{matrix}\, \right|\].   Let D be their product.   Then \[D=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\,\,\,\times \left| \,\begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} & {{\gamma }_{1}}  \\ {{\alpha }_{2}} & {{\beta }_{2}} & {{\gamma }_{2}}  \\ {{\alpha }_{3}} & {{\beta }_{3}} & {{\gamma }_{3}}  \\ \end{matrix}\, \right|\]     \[=\left| \,\begin{matrix} {{a}_{1}}{{\alpha }_{1}}+{{b}_{1}}{{\beta }_{1}}+{{c}_{1}}{{\gamma }_{1}} & {{a}_{1}}{{\alpha }_{2}}+{{b}_{1}}{{\beta }_{2}}+{{c}_{1}}{{\gamma }_{2}} & {{a}_{1}}{{\alpha }_{3}}+{{b}_{1}}{{\beta }_{3}}+{{c}_{1}}{{\gamma }_{3}}  \\ {{a}_{2}}{{\alpha }_{1}}+{{b}_{2}}{{\beta }_{1}}+{{c}_{2}}{{\gamma }_{1}} & {{a}_{2}}{{\alpha }_{2}}+{{b}_{2}}{{\beta more...

  (1) Differentiation of a determinant   (i) Let \[\Delta (x)\] be a determinant of order two. If we write \[\Delta (x)=|{{C}_{1}}\,\,\,\,\,{{C}_{2}}|\], where \[{{C}_{1}}\] and \[{{C}_{2}}\] denote the 1st and 2nd columns, then   \[\Delta '(x)=\left| \,\begin{matrix}  C{{'}_{1}} & {{C}_{2}}  \\ \end{matrix} \right|+\left| \,\begin{matrix} {{C}_{1}} & {{{{C}'}}_{2}}  \\ \end{matrix} \right|\]   where \[C{{'}_{i}}\] denotes the column which contains the derivative of all the functions in the \[{{i}^{th}}\]column \[{{C}_{i}}\].   In a similar fashion, if we write \[\Delta (x)=\left| \,\begin{matrix} {{R}_{1}}  \\ {{R}_{2}}  \\ \end{matrix}\, \right|\], then \[{\Delta }'\,(x)=\left| \,\begin{matrix} R{{'}_{1}}  \\ {{R}_{2}}  \\ \end{matrix}\, \right|\,+\,\left| \,\begin{matrix} {{R}_{1}}  \\ {{{{R}'}}_{2}}  \\ \end{matrix}\, \right|\,\]   (ii) Let \[\Delta (x)\] be a determinant of order three. If we write \[\Delta (x)=\left| \,\begin{matrix} {{C}_{1}} & {{C}_{2}} & {{C}_{3}}\,  \\ \end{matrix} \right|\], then     \[\Delta '(x)=\left| \,\begin{matrix} C{{'}_{1}} & {{C}_{2}} & {{C}_{3}}\,  \\ \end{matrix} \right|+\left| \,\begin{matrix} {{C}_{1}} & C{{'}_{2}} & {{C}_{3}}\,  \\ \end{matrix} \right|+\left| more...

  (1) Solution of system of linear equations in three variables by Cramer's rule : The solution of the system of linear equations  \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z={{d}_{1}}\]                       .....(i)   \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z={{d}_{2}}\]                       .....(ii)   \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z={{d}_{3}}\]                       .....(iii)   Is given by \[x=\frac{{{D}_{1}}}{D},\,\,\,\,\,\,y=\frac{{{D}_{2}}}{D}\] and \[z=\frac{{{D}_{3}}}{D}\],   where, \[D=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\,,\]        \[{{D}_{1}}=\left| \,\begin{matrix} {{d}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{d}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{d}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\]   \[{{D}_{2}}=\left| \,\begin{matrix} {{a}_{1}} & {{d}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{d}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{d}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\,,\] and \[{{D}_{3}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{d}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{d}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{d}_{3}}  \\ \end{matrix}\, \right|\]   Provided that \[D\ne 0\]   (2) Conditions for consistency : For a system of 3 simultaneous linear equations in more...

  (1) Symmetric determinant   A determinant is called symmetric determinant if for its every element \[{{a}_{ij}}\,=\,\,\,{{a}_{ji\,}}\forall \,\,i,\,j\] e.g., \[\left| \,\begin{matrix} a & h & g  \\ h & b & f  \\  g & f & c  \\ \end{matrix}\, \right|\].   (2) Skew-symmetric determinant : A determinant is called skew symmetric determinant if for its every element \[{{a}_{ij}}\,=\,-\,{{a}_{ji\,\,}}\forall \,i,\,j\] e.g.,  \[\left| \,\begin{matrix} 0 & 3 & -1  \\ -3 & 0 & 5  \\ 1 & -5 & 0  \\ \end{matrix}\, \right|\]  

• Every diagonal element of a skew symmetric determinant is always zero.

 

• The value of a skew symmetric determinant of even order is always a perfect square and that of odd order is always zero.

    (3) Cyclic order : If elements of the rows more...

 A rectangular arrangement of numbers (which may be real or complex numbers) in rows and columns, is called a matrix. This arrangement is enclosed by small ( ) or big [ ] brackets. The numbers are called the elements of the matrix or entries in the matrix.

A matrix having \[m\] rows and \[n\] columns is called a matrix of order \[m\times n\] or simply \[m\times n\] matrix (read as an \[m\] by \[n\] matrix). A matrix A of order \[m\times n\] is usually written in the following manner   \[A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} & ...{{a}_{1j}} & ...{{a}_{1n}}  \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} & ...{{a}_{2j}} & ...{{a}_{2n}}  \\ ..... & ..... & ..... & ..... & .....  \\ {{a}_{i1}} & {{a}_{i2}} & {{a}_{i3}} & ...{{a}_{ij}} & ...{{a}_{in}}  \\ ..... & ..... & ..... & ..... & .....  \\ {{a}_{m1}} & {{a}_{m2}} & {{a}_{m3}} & ...{{a}_{mj}} & ...{{a}_{mn}}  \\ \end{matrix} \right]\,\text{or }A={{[{{a}_{ij}}]}_{m\times n}}\],   where \[\begin{align} & i=1,\,\,2,.....m \\  & j=1,\,\,2,.....n \\  \end{align}\]   Here \[{{a}_{ij}}\]denotes the element of \[{{i}^{th}}\] row and \[{{j}^{th}}\] column.   Example : order of matrix \[\left[ \begin{matrix} 3 & -1 & 5  \\ 6 & 2 & -7  \\ more...

Two matrix A and B are said to be equal matrix if they are of same order and their corresponding elements are equal.