If \(\begin{vmatrix} 2x+5 & 3 \\ 5x+2 & 9 \end{vmatrix}=0\) find x.

If A=\(\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}\) write A^-1 in terms of A .

Find the value of X, such that the points (0, 2), (1, x) and (3, 1) are collinear.

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.

Evaluate x if: \(\left| \begin{matrix} 2 & 4 \\ 5 & 1 \end{matrix} \right| =\left| \begin{matrix} 2x & 4 \\ 6 & x \end{matrix} \right| \)

For what value of x, the given matrix \(A=\left| \begin{matrix} 3-2x & x+1 \\ 2 & 4 \end{matrix} \right| \) is a singular matrix?

For what value of x, the matrix \(\left| \begin{matrix} 1+x & 7 \\ 3-x & 8 \end{matrix} \right| \) is a singular matrix?

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

If A is square matrix of order 3 and | A | = 5, find the value of |-3,4|

Evaluate : \(\begin{vmatrix} 2 &4 \\-1 & 2 \end{vmatrix}\)

Evaluate the determinant \(\Delta =\left| \begin{matrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{matrix} \right| \)

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.

If A = \(\left| \begin{matrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{matrix} \right| \), find |A|

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

Using the properties of determinants, solve the following for 'x' \(\left| \begin{matrix} x+2 & x+6 & x-1 \\ x+6 & x-1 & x+2 \\ x-1 & x+2 & x+6 \end{matrix} \right| =0\)

Using properties of determinants solve the following for X:
\(\begin{vmatrix} x-a & x & x \\ x & x+a & x \\ x & x & x+a \end{vmatrix}=0,a\neq 0\)

A school wants to award its student or the values of Honesty, Regularity and Hard Work with a total cash award of Rs.6,000. Three times the award money for Hard work added to that given for Honesty amounts to Rs.11,000. The award money given for Honesty and Hard work together is double the one given for regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

Using properties of determinants, prove that:
\(\left| \begin{matrix} { (y+z) }^{ 2 } & xy & zx \\ xy & { (x+z) }^{ 2 } & yz \\ xz & yz & { (x+y) }^{ 2 } \end{matrix} \right| =2xyz{ (x+y+z) }^{ 3 }.\)

If \(A=\left[ \begin{matrix} 1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1 \end{matrix} \right] ,\) find A^-1. Hence solve the following system of equations:
x + 2y + 5z = 10, x - y - z = -2, 2x + 3y - z = -11.