Write the adjoint of the following matrix \(\begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}\)

If \(\begin{vmatrix} x+1 & x-1 \\ x-3 & x+2 \end{vmatrix}=\begin{vmatrix} 4 & -1 \\ 1 & 3 \end{vmatrix}\)then write the value of x.

A is a non-singular matrix of order 3 and |A|=-4. Find |adj A|

In the interval  \(\frac { \pi }{ 2 }\) , find the value of x for which that matrix \(\left[ \begin{matrix} 2sinx \\ 1 \end{matrix}\begin{matrix} 3 \\ 2sinx \end{matrix} \right] \) is singular.

If \(\left| \begin{matrix} 2x & 5 \\ 8 & x \end{matrix} \right| =\left| \begin{matrix} 6 & -2 \\ 7 & 3 \end{matrix} \right| \), Write the value of x.

If A is a \(3\times 3\) matrix, \(\left| A \right| \neq 0\) and \(\left| 3A \right| =k\left| A \right| \), then write the value of k.

Given  \(A=\left( \begin{matrix} 4 & 2 & 5 \\ 2 & 0 & 3 \\ -1 & 1 & 0 \end{matrix} \right) \),write the value of det. (2AA^-1)

If \(\left| \begin{matrix} 2x & x+3 \\ 2(x+1) & x+1 \end{matrix} \right| =\left| \begin{matrix} 1 & 5 \\ 3 & 3 \end{matrix} \right| \), write the value of x.

If \(\left| \begin{matrix} x+1 & x-1 \\ x-3 & x+2 \end{matrix} \right| =\left| \begin{matrix} 4 & -1 \\ 1 & 3 \end{matrix} \right| \), then write the value of x.

If A_jj is the cofactor of the element a_jj  of the determinant \(\left| \begin{matrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{matrix} \right| \), then write the value of a ₃₂A₃₂.

If \(A=\left| \begin{matrix} 3 & 10 \\ 2 & 7 \end{matrix} \right| \),  then write A^-1

Is A is square matrix of order 3 and |2A|=k|A|, then find the value of k.

If \(\Delta =\left| \begin{matrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{matrix} \right| \)  write the minor of the element a₃₂.

For what values of k, the system of linear equations
x+y+z=2
2x+y-z=3
3x+2y+kz=4
has a unique solution?

If \(A=\left| \begin{matrix} 2 & 3 & -1 \\ 4 & 1 & 0 \\ 3 & 3 & 2 \end{matrix} \right| \) , find \(M_{ 12 }\times M_{ 21 }+C_{ 21 }\times C_{ 12 }\) when M_jj called minor and C_jj called co-factors of A.

If \(A=\left| \begin{matrix} 6 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{matrix} \right| \), then show that |2A| = 8|A|:

Evaluate: \(\Delta=\left| \begin{matrix} 0 & sin\alpha & -cos\alpha \\ -sin\alpha & 0 & sin\beta \\ cos\alpha & -sin\beta & 0 \end{matrix} \right| \)

Find the co - factors of the elements of the determinant: \(\left| \begin{matrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{matrix} \right| \) and verify that a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃ = 0.

Using properties of determinants prove that \(\left|\begin{matrix}a&a^2&\beta+\gamma\\ \beta&\beta^2&\gamma+a\\ \gamma&\gamma^2&a+\beta\end{matrix}\right|=(\beta-\gamma)(\gamma-\alpha)(a-\beta)(a+\beta+\gamma).\)

Using properties of determinants prove that \(\left| \begin{matrix} 1 & 1+p & 1+p+q \\ 2 & 3+2p & 4+3p+2q \\ 3 & 6+3p & { 10+6p+3q } \end{matrix} \right| =1\)

If x, y, z are non-zero numbers, then the inverse of matrix \(A=\left| \begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix} \right| is:\)
\((A)=\left[ \begin{matrix} { x }^{ -1 } & 0 & 0 \\ 0 & y^{ -1 } & 0 \\ 0 & 0 & { z }^{ -1 } \end{matrix} \right] \)  \((B)xyz\left[ \begin{matrix} { x }^{ -1 } & 0 & 0 \\ 0 & y^{ -1 } & 0 \\ 0 & 0 & { z }^{ -1 } \end{matrix} \right] \)

\((C)\frac { 1 }{ xyz } \left[ \begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix} \right] \) \((D)\frac { 1 }{ xyz } \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \)

If \(\Delta =\left| \begin{matrix} 1 & x & x^{ 2 } \\ 1 & y & { y }^{ 2 } \\ 1 & z & { z }^{ 2 } \end{matrix} \right| ,{ \Delta }_{ 1 }=\left| \begin{matrix} 1 & 1 & 1 \\ yz & zx & xy \\ x & y & z \end{matrix} \right| \)then prove that \(\Delta+\Delta_1=0\)

IN a triangle ABC, if:\(\left| \begin{matrix} 1 & 1 & 1 \\ 1+sinA & 1+sinB & 1+sinC \\ sinA+sin^{ 2 }A & sinB+sin^{ 2 }B & sinC+sin^{ 2 }C \end{matrix} \right| =0\) then prove \(\Delta\) ABC ia an isosceles triangle

If\(\Delta=\left|\begin{matrix}0&b-a&c-a\\ a-b&0&c-b\\a-c&b-c&0\end{matrix}\right|\), then show that \(\Delta\) is equal to zero

If a, b, c are different, then the determinant : \(\left| \begin{matrix} 1 & 1 & { 1 } \\ { (x- }a)^{ 2 } & (x-b)^{ 2 } & { (x-c) }^{ 2 } \\ (x-b)(x-c) & (x-c)(x-a) & (x-a)(x-b) \end{matrix} \right| \)
vanishes when \(x={1\over 3}(a+b+c)\)

Using properties of determinants, show that triangle ABC is isosceles if: 
\(\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0\)