Determinants

### Mark the tick against the correct answer in the following: A. 0 B. 1 C. D. none of these

Solution: Option(B) To find: Value of $\left|\begin{array}{ccc}1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q\end{array}\right|$ We have,...

### Choose the correct answer Let A =(Fig 1) where Then:

Fig 1= (A) Det (A) = 0  (B) Det (A)    (C) Det (A)   (D) Det (A)  SOLUTION: Given: Matrix A =        ……….(i) Since      [  cannot be negative]       Therefore, The correct option is option...

### Choose the correct answer If x,y,z are non-zero real numbers, then the inverse of matrix A =(fig1) is

(A)  (B)   (C)  (D)   SOLUTION: Given: Matrix A =        exists and unique solution is  ……….(i) Now     and  and   adj. A =  =  And  =  =  =  Therefore, option (A) is...

### Choose the correct answer in If a,b,c are in A.P., then the determinant (fig 1) is:

fig 1: (A) 0   (B) 1   (C)      (D)   solution: According to question,  ……….(i) Let  =   =  [From eq. (i)] = 0 [ R2 and R3 have become identical] Therefore, option (A) is...

### Solve the system of the following equations: (Using matrices):

SOLUTION: Putting  and  in the given equations,     the matrix form of given equations is  [AX= B] Here,   A =  X =  and B =    =  =    exists and unique solution is  ……….(i) Now     and  and   adj....

### Using properties of determinants in , prove that:

=0 SOLUTION: L.H.S. =  =  =   =  =  =  [ C2 and C3 have become identical] = 0 = R.H.S.

### Using properties of determinants in , prove that:

SOLUTION: L.H.S. =  =   =  =  =  = 1 = R.H.S.

### Using properties of determinants in , prove that:

SOLUTION: L.H.S. =  =   =  =   =  =  =  =  =  = R.H.S.

### Using properties of determinants in , prove that:

SOLUTION:  L.H.S. =  =  =  (say) ……….(i) Now  =  =  =   From eq. (i), L.H.S. =  ……….(ii) Now  =   Expanding along third column,  =  =  =  =  =   From eq. (i), L.H.S. =  =  =...

### Using properties of determinants in , prove that:

SOLUTION: L.H.S. =  =   = =   Expanding along third column,  =  =  =  =  =  =  = R.H.S.

### Evaluate:

SOLUTION: Let  =   =  =  =

### Evaluate:

SOLUTION:  Let  =   =  =   =  =  =  =  =  =

### Let A=(fig 1), prove that (i) (adj A)^-1 = adj(A)^-1 (ii) (A^-1)^-1=A

Given: Matrix A =      =  Therefore,  exists.   and  and   adj. A =  = B (say)   =            ………(i)   =  =  Therefore,  exists.   and  and   adj. B =  =    =  =  ….(ii) Now to find  (say), where C...

### If A^-1 and B (in fig 1) is given , find (AB)^-1.

fig 1: ,   B= solution: Given:  and B =  Since,   [Reversal law] ……….(i) Now  =  =  Therefore,  exists.   and  and   adj. B =  =    From eq. (i),   ...

### Prove that : fig 1

fig 1: solution: L.H.S. =  =  =  =   =  =   =  =  = R.H.S.    Proved.

### Solve the equation: fig1

fig1:      Either    ……….(i) Or           But this is contrary as given that . Therefore, from eq. (i),  is only the solution.

### If a, b and c are real numbers, and fig 1 Show that either a+b+c=0 or a=b=c

fig 1=  Given:       Here,   Either   ……….(i) Or       [Expanding along first row]                 and  and     and  and  ……….(ii) Therefore, from eq. (i) and (ii),...

### Evaluate: fig1

fig 1: Expanding along first row, = =  =  =  = 1

### Without expanding the determinant, prove that

LHS: $\left|\begin{array}{lll}a & a^{2} & b c \\ b & b^{2} & c a \\ c & c^{2} & a b\end{array}\right|$ Multiplying R 1 by a $\mathrm{R} 2$ by $\mathrm{b}$ and $\mathrm{R} 3$...

### Examine the consistency of the system of equations in & &

Given set of equations is: $5 x-y+4 z=5 ; 2 x+3 y+5 z=2 ; 5 x-2 y+6 z=-1$ This set of equation in the form of matrix is $A X=B$ $=140-13-76$ $=140-89$ $=51$ $\neq 0$ System of equations is...

### Let and verify that

$A=\left[\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right]$ $|\mathrm{A}|=\left|\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right|=1 \neq 0$ $\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|}$...

### Find the inverse of each of the matrices (if it exists) given in

Let A =    =    exists. A11 = , A12 = , A13 = , A21 = ,   A22 = , A23 = ,  A31 = , A32 = , A33 =   adj. A =

### Find the inverse of each of the matrices (if it exists) given in

Let A =    =    exists. A11 = ,  A12 = , A13 = , A21 = , A22 = , A23 = , A31 = , A32 = , A33 =   adj. A =

### Find the inverse of each of the matrices (if it exists) given in

Let $\mathrm{A}=\left[\begin{array}{ccc}2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1\end{array}\right]$ Therefore, $A^{-1}$ exists A11 = , A12 = , A13 = , A21 = , A22 = , A23 = ,...

### Find the inverse of each of the matrices (if it exists) given in

Let $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$ Therefore, $A^{-1}$ exists Find adj A:  212.  \text { 212. }...

### Find adjoint of each of the matrices in

Let $A=$ $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ Cofactors of the above matrix are 11 $A_{11}=4$ $A_{12}=-3$ $\mathrm{A}_{21}=-2$ $A_{22}=1$ adj....

### Using Cofactors of elements of third column, evaluate .

Elements of third column of  are  A13 = Cofactor of  =  A23 = Cofactor of  =  A33 = Cofactor of  =    =  =  =  =  =  =  =  =  =

### Using Cofactors of elements of second row, evaluate

Find Cofactors of elements of second row: $A_{21}=$ Cofactor of element $a_{21}=(-1)^{2+1}\left|\begin{array}{cc}3 & 8 \\ 2 & 3\end{array}\right|-(-1)^{3}(9-16)=7$ $A_{22}=C$ ofactor of...

### Write Minors and Cofactors of the elements of following determinants: (i)(ii)

(i) $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$ Find Minors and cofactors of elements: Say, $M_{i j}$ is minor of element $a_{i j}$ and...

### Write Minors and Cofactors of the elements of following determinants: (i) (ii)

Find Minors of elements: Say, $M_{i j}$ is minor of element $a_{i i}$ $\mathrm{M}_{11}=$ Minor of element $\mathrm{a}_{11}=3$ $\mathrm{M}_{12}=$ Minor of element $\mathrm{a}_{12}=0$...

### If area of triangle is 35 sq units with vertices and Then is (A) 12 (B) (C) (D)

The correct option is OPTION(D) Given: Area of triangle = Modulus of  = 35  Modulus of  = 35           Taking positive sign,    Taking negative sign,

### (i) Find equation of line joining and using determinants. (ii) Find equation of line joining and using determinants.

Let $A(x, y)$ be any vertex of a triangle. All points are on one line if area of triangle is zero. $\frac{1}{2}[x(2-6)-y(1-3)+1(6-6)]=0$ $-4 x+2 y=0$ $y=2 x$ Which is equation of line. (ii) Let...

### Find area of the triangle with vertices at the point given in each of the following:(iii)

Area $=\frac{1}{2}\left|\begin{array}{ccc}-2 & -3 & 1 \\ 3 & 2 & 1 \\ -1 & -8 & 1\end{array}\right|$ $=\frac{1}{2}[-2(10)+3(4)-22]$ $=15$ sq. Units

### Find area of the triangle with vertices at the point given in each of the following: (i) (ii)

Formula for Area of triangle: $\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right|$ (i)...

### Choose the corrrect answer Which of the following is correct (A) Determinant is a square matrix. (B) Determinant is a number associated to a matrix. (C) Determinant is a number associated to a square matrix. (D) None of these

The correct option is Option (C) Since, Determinant is a number associated to a square matrix.

### Choose the corrrect answer Let be a square matrix of order , then is equal to (A) (C) (D)

The correct option is Option (C). Let A =  be a square matrix of order 3 x 3.   ……….(i)          =  [From eq. (i)]

### By using properties of determinants, show that:

LHS $\left|\begin{array}{ccc}a^{2}+1 & a b & a c \\ a b & b^{2}+1 & b c \\ c a & c b & c^{2}+1\end{array}\right|$ Multiplying  by  respectively and then dividing the...

### By using properties of determinants, show that:

(i) LHS =  [operating  and ] =  =  = R.H.S. (ii) L.H.S. =  =   =  =  [operating  and ] =  =  =  = R.H.S.  Proved.

### By using properties of determinants, show that:

LHS= $\left|\begin{array}{lll}x & x^{2} & y z \\ y & y^{2} & z x \\ z & z^{2} & x y\end{array}\right|$ Mulitiplying $R_{1}, R_{2}, R_{3}$ by $x, y, z$ respectively...

### By using properties of determinants, show that:(i) (ii)

(i) LHS: $\left|\begin{array}{lll}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right|$ $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ and...

### Using the properties of determinants and without expanding, prove that

Solving L.H.S $\left| \begin{matrix} -{{a}^{2}} & ab & ac \\ ba & -{{b}^{2}} & bc \\ ca & cb & -{{c}^{2}} \\ \end{matrix} \right|$ Taking a common from${{R}_{1}}$,b common...

### Using the properties of determinants and without expanding, prove that

Let $\Delta=\left|\begin{array}{ccc}0 & a & -b \\ -a & 0 & -c \\ b & c & 0\end{array}\right|$ Taking (-1) common from all the 3 rows. Again, interchanging rows and columns,...

### Using the properties of determinants and without expanding, prove that

LHS: Applying: $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ $\left|\begin{array}{ccc}b+c+c+a+a+b & q+r+r+p+p+q & y+z+z+x+x+y \\ c+a & r+p & z+x \\ a+b & p+q & x+y\end{array}\right|$...

### Using the properties of determinants and without expanding, prove that:

$\left|\begin{array}{ccc}1 & b c & a(b+c) \\ 1 & c a & b(c+a) \\ 1 & a b & c(a+b)\end{array}\right|$ Applying: $\mathrm{C}_{3}->\mathrm{C}_{3}+\mathrm{C}_{2}$...

### Evaluate the determinants(iii) (iv)

(iii) = 0 + 6 – 6 = 0   (iv) =5

$A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]$ $2 \mathrm{~A}=\left[\begin{array}{ll}2 & 4 \\ 8 & 4\end{array}\right]$ L.H.S. $=|2 A|=\left|\begin{array}{ll}2 & 4 \\... read more ### Evaluate the following determinants & (i)$\left| \begin{matrix} \cos \theta  & -\sin \theta   \\ \sin \theta  & \cos \theta   \\ \end{matrix} \right|$=$\cos \theta X\cos \theta -(-\sin \theta )Xsin\theta =1$(ii)$\left|...
$\left|\begin{array}{cc}2 & 4 \\ -5 & -1\end{array}\right|=2(-1)-4(-5)=18$