If A is a square matrix of order 3 and |3A| = k|A|, then write the value of k.

What positive value of x makes the following pair of determinants equal?
\(\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}\)

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

Evaluate \(\begin{vmatrix} cos15^{ o } & sin15^{ o } \\ sin75^{ o } & cos75^{ o } \end{vmatrix}\)

Given determinant \(\begin{vmatrix} a_{ 11 } & a_{ 12 } & a_{ 13 } \\ a_{ 21 } & a_{ 22 } & a_{ 23 } \\ a_{ 31 } & a_{ 32 } & a_{ 33 } \end{vmatrix}\).
Find the value of a₁₁A₂₁+a₁₂A₂₂+a₁₃A₂₃, where A_ij is cofactor of element a_ij

Find the Value of determinant \(\triangle =\left| \begin{matrix} 2 & 2 & 2 \\ x & y & z \\ y+z & z+x & z+y \end{matrix} \right| \)

For what value of k, the matrix \(\left[ \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right] \)has no inverse

Let Abe a non singular matrix of order 3 x 3 , such that |AdjA| = 100 find |A|

If A is non singular matrix of order n, then weite the value of Adj(Adj A) and hence write the value of Adj(Adj A) if order of A and |A| = 5

Let A be a diagonal A = (d₁, d₂, …, dn ) write the value 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: \(\Delta=\left| \begin{matrix} 0 & sin\alpha & -cos\alpha \\ -sin\alpha & 0 & sin\beta \\ cos\alpha & -sin\beta & 0 \end{matrix} \right| \)

Show that the matrix \(A=\left| \begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix} \right| \)satisfies the equation A²- 4A + I = 0, where I is 2 x 2 identity matrix and O is 2 x 2 zero matix. Using this equation, find A^–1.

Find the values of x, if:
(i)\(\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}\)
(ii)\(\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix}=\begin{vmatrix} x & 3 \\2x & 5 \end{vmatrix}\)

Prove that : \(\left| \begin{matrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2ab \end{matrix} \right| =2(a+b+c{ ) }^{ 3 }\)

If a,b,c are positive and unequal, show that the value of the determinant \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\) is negative.

\(\left| \begin{matrix} { yz }-x^{ 2 } & { zx }-y^{ 2 } & { xy-z }^{ 2 } \\ { zx-y }^{ 2 } & { xy }-z^{ 2 } & { yz-x }^{ 2 } \\ { xy-z }^{ 2 } & { yz }-x^{ 2 } & { zx-x }^{ 2 } \end{matrix} \right| \)is divisible by (x+y+z) and here, find the quotient.

A(adj A) = (adj A)A = |A| I for matrix \(A=\left| \begin{matrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{matrix} \right| .\)

Using matrix method, determine whether the following system of equation is consisten or inconsistent
3x - y -2z =2
2y -z = -1
3x - 5y = 3