List of top Mathematics Questions on Determinants

Obtain the value of \[ \Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix} \] in terms of \(x, y, z\). Further, if \(\Delta = 0\) and \(x, y, z\) are non–zero real numbers, prove that \[ x^{-1} + y^{-1} + z^{-1} = -1 \]

If \[ \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & 4 \\ 0 & 4 & 2a \end{vmatrix} = -86 \] then the sum of all possible values of \( a \) is

Let \( A \) be a square matrix of order 3 and \( |A| \) denote the determinant of \( A \). If \( A = kB \), where \( k \) is a scalar, then \( |A| \) is equal to

The value of \( k \) for which the system of equations \( x + ky + 3z = 0 \); \( 3x + ky + 2z = 0 \); \( 2x + 3y + 4z = 0 \) has a non-trivial solution is

If \( A = \text{diag}(d_1, d_2, \ldots, d_n) \), then \( |A| \) is:

Let \( A \) be a square matrix of order 3x3. Then \( |2A| \) is equal to:

Using determinants, find the value of \( k \) if the area of the triangle formed by the points \( (-3, 6) \), \( (-4, 4) \), and \( (k, -2) \) is 12 sq. units.

Prove that: \[ \left| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} \right| = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \]

If \[ \Delta = \begin{vmatrix} 3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3 \end{vmatrix} \] then find the value of $|\Delta|$.

Prove that \[ \begin{vmatrix} x & x^{2} & yz \\ y & y^{2} & zx \\ z & z^{2} & xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx). \]

Prove that \[ \begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} = 2(a+b+c)^3 \]

Let \( M \) and \( N \) be any two \( 4 \times 4 \) matrices with integer entries satisfying

\[ MN = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} = 2 \]

Then the maximum value of \( \det(M) + \det(N) \) is ...............

Using determinants, find the value of \( k \) if the area of the triangle formed by the points \( (-3, 6), (-4, 4) \) and \( (k, -2) \) is 12 sq. units.

If \( A \) is a square matrix of order \( 2 \times 2 \) and \( |A| = 5 \), then \( |Adj(A)| \) is: