$\left|\begin{array}{cc}x^2-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

$\left|\begin{array}{ccc}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{array}\right|$

$\left|\begin{array}{ccc}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{array}\right|$

$\left|\begin{array}{ccc}3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z\end{array}\right|$

$\left|\begin{array}{ccc}x+4 & x & x \\ x & x+4 & x \\ x & x & x+4\end{array}\right|$

$\left|\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\right|$

$\left|\begin{array}{ccc}y^2 z^2 & y z & y+z \\ z^2 x^2 & z x & z+x \\ x^2 y^2 & x y & x+y\end{array}\right|=0$

$\left|\begin{array}{ccc}y+z & z & y \\ z & z+x & x \\ y & x & x+y\end{array}\right|=4 x y z$

$\left|\begin{array}{ccc}a^2+2 a & 2 a+1 & 1 \\ 2 a+1 & a+2 & 1 \\ 3 & 3 & 1\end{array}\right|=(a-1)^3$

If $A+B+C=0$, then prove that $\left|\begin{array}{ccc}1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1\end{array}\right|=0$.

If the coordinates of the vertices of an equilateral triangle with sides of length ' $a$ ' are $\left(x_1, y_1\right),\left(x_2, y_2\right)$ and $\left(x_3, y_3\right)$, then

$$\left|\begin{array}{lll} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array}\right|^2=\frac{3 a^4}{4}$$

Find the value of $\theta$ satisfying $\left[\begin{array}{rrc}1 & 1 & \sin 3 \theta \\ -4 & 3 & \cos 2 \theta \\ 7 & -7 & -2\end{array}\right]=0$

If $\left[\begin{array}{lll}4-x & 4+x & 4+x \\ 4+x & 4-x & 4+x \\ 4+x & 4+x & 4-x\end{array}\right]=0$, then find the value of $x$.

If $a_1, a_2, a_3, \ldots, a_r$ are in GP, then prove that the determinant $\left|\begin{array}{ccc}a_{r+1} & a_{r+5} & a_{r+9} \\ a_{r+7} & a_{r+11} & a_{r+15} \\ a_{r+11} & a_{r+17} & a_{r+21}\end{array}\right|$ is independent of $r$.

Show that the points $(a+5, a-4),(a-2, a+3)$ and $(a, a)$ do not lie on a straight line for any value of $a$.

Show that $\triangle A B C$ is an isosceles triangle, if the determinant $$ \Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1+\cos A & 1+\cos B & 1+\cos C \\ \cos ^2 A+\cos A & \cos ^2 B+\cos B & \cos ^2 C+\cos C \end{array}\right|=0 .$$

Find $A^{-1}$, if $A=\left|\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right|$ and show that $A^{-1}=\frac{A^2-3 I}{2}$.

If $A=\left|\begin{array}{rrr}1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1\end{array}\right|$, then find the value of $A^{-1}$. Using $A^{-1}$, solve the system of linear equations $x-2 y=10$, $2 x-y-z=8$ and $-2 y+z=7$.

Using matrix method, solve the system of equations $3 x+2 y-2 z=3$, $x+2 y+3 z=6$ and $2 x-y+z=2$.

If $A=\left|\begin{array}{rrr}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right|$ and $B=\left|\begin{array}{rcr}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right|$, then find $B A$ and use this to solve the system of equations $y+2 z=7, x-y=3$ and $2 x+3 y+4 z=17$.

If $a+b+c \neq 0$ and $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$, then prove that $a=b=c$.

Prove that $\left|\begin{array}{lll}b c-a^2 & c a-b^2 & a b-c^2 \\ c a-b^2 & a b-c^2 & b c-a^2 \\ a b-c^2 & b c-a^2 & c a-b^2\end{array}\right|$ is divisible by $(a+b+c)$ and find the quotient.

If $x+y+z=0$, then prove that $\left|\begin{array}{lll}x a & y b & z c \\ y c & z a & x b \\ z b & x c & y a\end{array}\right|=x y z\left|\begin{array}{lll}a & b & c \\ c & a & b \\ b & c & a\end{array}\right|$.

If $\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}6 & -2 \\ 7 & 3\end{array}\right|$, then the value of $x$ is

The value of $\left|\begin{array}{lll}a-b & b+c & a \\ b-a & c+a & b \\ c-a & a+b & c\end{array}\right|$ is

If the area of a triangle with vertices $(-3,0),(3,0)$ and $(0, k)$ is 9 sq units. Then, the value of $k$ will be

The determinant $\left|\begin{array}{lll}b^2-a b & b-c & b c-a c \\ a b-a^2 & a-b & b^2-a b \\ b c-a c & c-a & a b-a^2\end{array}\right|$ equals to

The number of distinct real roots of $\left|\begin{array}{lll}\cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

If $A, B$ and $C$ are angles of a triangle, then the determinant $\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1\end{array}\right|$ is equal to

If $f(t)=\left[\begin{array}{ccc}\cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t\end{array}\right]$, then $\lim _\limits{t \rightarrow 0} \frac{f(t)}{t^2}$ is equal to

The maximum value of $\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1+\cos \theta & 1 & 1\end{array}\right|$ is (where, $\theta$ is real number)

If $f(x)=\left|\begin{array}{ccc}0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|$, then

If $A=\left|\begin{array}{rrr}2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3\end{array}\right|$, then $A^{-1}$ exists, if

If $A$ and $B$ are invertible matrices, then which of the following is not correct?

If $x, y$ and $z$ are all different from zero and $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z\end{array}\right|=0$, then the value of $x^{-1}+y^{-1}+z^{-1}$ is

The value of $\left|\begin{array}{ccc}x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x\end{array}\right|$ is

If $A$ is a matrix of order $3 \times 3$, then $|3 A|$ is equal to ............. .

If $A$ is invertible matrix of order $3 \times 3$, then $\left|A^{-1}\right|$ is equal to ............. .

If $x, y, z \in R$, then the value of $\left|\begin{array}{lll}\left(2^x+2^{-x}\right)^2 & \left(2^x-2^{-x}\right)^2 & 1 \\ \left(3^x+3^{-x}\right)^2 & \left(3^x-3^{-x}\right)^2 & 1 \\ \left(4^x+4^{-x}\right)^2 & \left(4^x-4^{-x}\right)^2 & 1\end{array}\right|$ is

If $\cos 2 \theta=0$, then $\left|\begin{array}{ccc}0 & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & 0 \\ \sin \theta & 0 & \cos \theta\end{array}\right|^2$ is equal to ............. .

If $A$ is a matrix of order $3 \times 3$, then $\left(A^2\right)^{-1}$ is equal to ............ .

If $A$ is a matrix of order $3 \times 3$, then $\left(A^2\right)^{-1}$ is equal to .............. .

If $A$ is a matrix of order $3 \times 3$, then the number of minors in determinant of $A$ are .......... .

The sum of products of elements of any row with the cofactors of corresponding elements is equal to ............ .

If $x=-9$ is a root of $\left|\begin{array}{lll}x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x\end{array}\right|=0$, then other two roots are ............ .

$\left|\begin{array}{ccc}0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{array}\right|$ is equal to ............ .

$$\text { If } \begin{aligned} f(x) & =\left|\begin{array}{ccc} (1+x)^{17} & (1+x)^{19} & (1+x)^{23} \\ (1+x)^{23} & (1+x)^{29} & (1+x)^{34} \\ (1+x)^{41} & (1+x)^{43} & (1+x)^{47} \end{array}\right| \\ & =A+B x+C x^2+\ldots, \text { then } A \text { is equal to ............. .} \end{aligned}$$

$\left(A^3\right)^{-1}=\left(A^{-1}\right)^3$, where $A$ is a square matrix and $|A| \neq 0$.

$(a A)^{-1}=\frac{1}{a} A^{-1}$, where $a$ is any real number and $A$ is a square matrix.

$\left|A^{-1}\right| \neq|A|^{-1}$, where $A$ is a non-singular matrix.

If $A$ and $B$ are matrices of order 3 and $|A|=5,|B|=3$, then $|3 A B|=27 \times 5 \times 3=405$.

If the value of a third order determinant is 12 , then the value of the determinant formed by replacing each element by its cofactor will be 144.

$\left|\begin{array}{lll}x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{array}\right|=0$, where $a, b$ and $c$ are in AP.

$|\operatorname{adj} A|=|A|^2$, where $A$ is a square matrix of order two.

The determinant $\left|\begin{array}{lll}\sin A & \cos A & \sin A+\cos B \\ \sin B & \cos A & \sin B+\cos B \\ \sin C & \cos A & \sin C+\cos B\end{array}\right|$ is equal to zero.

If the determinant $\left|\begin{array}{lll}x+a & p+u & l+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h\end{array}\right|$ splits into exactly $k$ determinants of order 3, each element of which contains only one term, then the value of $k$ is 8 .

If $\Delta=\left|\begin{array}{lll}a & p & x \\ b & q & y \\ c & r & z\end{array}\right|=16$, then $\Delta_1=\left|\begin{array}{lll}p+x & a+x & a+p \\ q+y & b+y & b+q \\ r+z & c+z & c+r\end{array}\right|=32$.

The maximum value of $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cos \theta\end{array}\right|$ is $\frac{1}{2}$.