List of top Mathematics Questions on Matrices and Determinants

If \( A \) and \( B \) are matrices and \( AB = BA = A^{-1} \) then the value of \( (A + B)(A - B) \) is
The matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} \), then adj \( (A) \) is equal to
If \( A = \begin{pmatrix} 1 & 2
3 & 1 \end{pmatrix} \), then rank \( A \) is
If \[ A = \begin{bmatrix} 3 & 4 \\ 5 & 7 \end{bmatrix}, \] then \( A \cdot \text{(adj A)} \) is equal to:
If the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are collinear, then the rank of the matrix \[ \begin{bmatrix} x_1 & y_1 & 1
x_2 & y_2 & 1
x_3 & y_3 & 1 \end{bmatrix} \]
The value of the determinant \[ \begin{vmatrix} \cos(\alpha - \beta) & \cos \alpha & \cos \beta
\cos(\alpha - \beta) & 1 & \cos \beta
\cos \alpha & \cos \beta & 1 \end{vmatrix} \]
If \( a \), \( b \), and \( c \) are in AP, then determinant \[ \begin{vmatrix} x+2 & x+3 & x+4
x+4 & x+5 & x+6
x+7 & x+8 & x+9 \end{vmatrix} \]
If \[ A = \begin{pmatrix} 1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9 \end{pmatrix}, \] then the trace of matrix \( A \) is:
If \[ \mathbf{I} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \] then \( B \) is equal to:
If \(B\) is a non-singular matrix and \(A\) is a square matrix, then \(\det(B^{-1}AB)\) is equal to
If \(f(x), g(x)\) and \(h(x)\) are three polynomials of degree 2 and
\[ \Delta(x)= \begin{vmatrix} f(x) & g(x) & h(x)\\ f'(x) & g'(x) & h'(x)\\ f''(x) & g''(x) & h''(x) \end{vmatrix} \] then \(\Delta(x)\) is a polynomial of degree
If \(x,y,z\) are different from zero and
\[ \Delta= \begin{vmatrix} a & b-y & c-z \\ a-x & b & c-z \\ a-x & b-y & c \end{vmatrix}=0 \] then the value of the expression \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\) is
The system of equations
\[ x+y+z=0 \] \[ 2x+3y+z=0 \] \[ x+2y=0 \] has
If \(\begin{vmatrix} 0 & a^4 \\ b & 0 \end{vmatrix}=1\), then
If \(D=\text{diag}(d_1,d_2,\ldots,d_n)\), where \(d_i\neq 0\), for \(i=1,2,\ldots,n\), then \(D^{-1}\) is equal to
If \( A(\theta) = \left[ \begin{matrix} 1 & -\tan \theta \\ \end{matrix} \right] \) and \( AB = 1 \), then \[ (\cos \theta)B \] is equal to
If \( x = -5 \) is a root of the equation \[ \begin{vmatrix} 2x+1 & 4 & 8 \\ 2 & 2x & 2 \\ 7 & 6 & 2x \end{vmatrix} = 0 \] then the other roots are
If the rank of the matrix \[ \begin{pmatrix} -1 & 2 & 5 \\ 2 & -4 & -4 \\ 1 & -2 & a + 1 \end{pmatrix} \] is 1, then the value of \( a \) is
The simultaneous equations \( Kx + 2y = 1 \), \( K(1 - y) - 2x = 2 \) and \( K + 2y = 3 \) have only one solution when
If \( x = -9 \) is a root of \( \begin{pmatrix} 2 & 3 \\ 7 & 6 \end{pmatrix} \times \begin{pmatrix} x \end{pmatrix} = 0 \), then other two roots are:
Let \( A = \begin{pmatrix} 1 & 3 & 2 \\ 4 & 2 & 5 \\ 7 & -t & -6 \end{pmatrix} \), then the values of \( t \) for which inverse of \( A \) does not exist are:
The value of \( x \), for which the matrix \( A \) is singular, is: \[ A = \begin{pmatrix} 2 & x & -1 & 2 \\ 1 & x & 2x^2 \\ 1 & \frac{1}{x} & 2 \end{pmatrix} \]
The values of \( \alpha \) for which the system of equation \( x + y + z = 1 \), \( x + 2y + 4z = \alpha \), \( x + 4y + 10z = \alpha^2 \) is consistent are given by: