List of top Mathematics Questions on Matrices and Determinants asked in AP EAPCET

Let \( A = \begin{bmatrix} 1 & 2 2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} x & y 1 & 2 \end{bmatrix} \) be two matrices such that \( (A + B)(A - B) = A^2 - B^2 \).
If \( C = \begin{bmatrix} x & 2 1 & y \end{bmatrix} \), then trace \((C) = \)

If \[ A= \begin{bmatrix} b+c & a & a\\ b & c+a & b\\ c & c & a+b \end{bmatrix} \] is a matrix such that trace of $A=18$ and \[ \det(A)=96, \] if $a,b,c\in \mathbb{N}$ and $ab=6$, then $ab+bc+ca=$

Let \(S\) be a symmetric matrix obtained from \[ A= \begin{bmatrix} 1 & 2 & -3\\ 2 & -2 & 1\\ 3 & 1 & -1 \end{bmatrix} \] and \(T\) be a skew-symmetric matrix obtained from \[ B= \begin{bmatrix} 4 & 2 & 0\\ 1 & -1 & 3\\ 0 & 2 & -3 \end{bmatrix} \] If trace of \(S=-4\) and the non-zero elements of \(T\) are \(-1,1\), then \(S+T=\)

For a \( 3 \times 3 \) non singular matrix A, if \( \text{Adj}(\text{Adj}(\text{Adj}(\text{Adj}(A)))) = |A|^n A \), then \( n = \)

On a matrix A when three elementary operations namely interchange of \( R_1 \) and \( R_2 \), \( R_2 \rightarrow R_2 - 2R_1 \), \( R_3 \rightarrow R_3 - 3R_1 \) are applied successively, A is transformed to \( \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix} \). Then \( \text{Tr}(A) = \)

If \( A=\begin{bmatrix}1&-2&2\\ 2&1&-2\\ 2&K&4\end{bmatrix} \), \( B=\begin{bmatrix}2&4&3\\ 3&4&5\\ 1&2&2\end{bmatrix} \) and \( \text{Rank}(A)=2 \), then \( K + \text{Rank}(B) = \)

Let \( A, B \) be \( 3^{\text{rd}} \) order non-singular square matrices and \( K \) is a real number. Which of the following is true?

The augmented matrix of a non-homogeneous system of equations \( AX = B \) is reduced to the following form after applying a series of elementary row transformations: \[ \begin{bmatrix}1&1&1&5\\ 0&0&-3&4\\ 0&0&\mu+1&\lambda^{2}-2\lambda+1\end{bmatrix} \] Then, which of the following is correct?

If \[ y= \begin{vmatrix} \sin x & \cos x & \sin x\\ \cos x & -\sin x & \cos x\\ x & 1 & 1 \end{vmatrix}, \] then \[ \frac{dy}{dx}= \]

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+bx+c=0$, then \[ \begin{vmatrix} \alpha & \beta & \gamma\\ \beta & \gamma & \alpha\\ \gamma & \alpha & \beta \end{vmatrix} = \]

For a fixed positive integer $n$, if \[ D= \begin{vmatrix} n! & (n+1)! & (n+2)!\\ (n+1)! & (n+2)! & (n+3)!\\ (n+2)! & (n+3)! & (n+4)! \end{vmatrix}, \] then \[ \frac{D}{n!(n+1)!(n+2)!} = \]

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A^{-1} =$

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, then $A^2 - 5A - 2I =$

If \( a \neq b \neq c \), \( \Delta_1 = \begin{vmatrix} 1 & a^2 & bc 1 & b^2 & ca 1 & c^2 & ab \end{vmatrix} \), \( \Delta_2 = \begin{vmatrix} 1 & 1 & 1 a^2 & b^2 & c^2 a^3 & b^3 & c^3 \end{vmatrix} \) and \( \frac{\Delta_1}{\Delta_2} = \frac{6}{11} \), then \( 11(a + b + c) = \)

If \( \begin{vmatrix} 9 & 25 & 16 16 & 36 & 25 25 & 49 & 36 \end{vmatrix} = K \), then \( K, K + 1 \) are the roots of the equation:

If \( A = \begin{bmatrix} 5 & 5\alpha & \alpha 0 & \alpha & 5\alpha 0 & 0 & 5 \end{bmatrix} \) and \( \det(A^2) = 25 \), then \( |\alpha| = \)

\( A = \begin{bmatrix} 0 & k & k k & -4 & -6 k & -3 & -5 \end{bmatrix} \) is a singular matrix for:

The determinant \[ \det \begin{bmatrix} \frac{a^2 + b^2}{c} & c & c a & \frac{b^2 + c^2}{a} & a b & b & \frac{c^2 + a^2}{b} \end{bmatrix} \] is equal to:

If \( A = \begin{bmatrix} x & 2 & 1 -2 & y & 0 2 & 0 & -1 \end{bmatrix} \), \( x \) and \( y \) are non-zero numbers, trace of \( A = 0 \) and determinant of \( A = -6 \), then the minor of the element \( 1 \) of \( A \) is:

If \( \begin{vmatrix} 1 & 2 & 3 - \lambda 0 & -1 - \lambda & 2 1 - \lambda & 1 & 3 \end{vmatrix} = A\lambda^3 + B\lambda^2 + C\lambda + D \), then \( D + A = \)

A value of \( \theta \) lying between \( 0 \) and \( \pi / 2 \) and satisfying \[ \begin{vmatrix} 1 + \sin^2 \theta & \cos^2 \theta & 4 \sin 4\theta \sin^2 \theta & 1 + \cos^2 \theta & 4 \sin 4\theta \sin^2 \theta & \cos^2 \theta & 1 + 4 \sin 4\theta \end{vmatrix} = 0 \] is:

If \( A = \begin{bmatrix} x & y & y y & x & y y & y & x \end{bmatrix} \) is a matrix such that \( 5A^{-1} = \begin{bmatrix} -3 & 2 & 2 2 & -3 & 2 2 & 2 & -3 \end{bmatrix} \), then \( A^2 - 4A = \)

If the system of simultaneous linear equations \( x + \lambda y - 2z = 1 \), \( x - y + \lambda z = 2 \) and \( x - 2y + 3z = 3 \) is inconsistent for \( \lambda = \lambda_1 \) and \( \lambda_2 \), then \( \lambda_1 + \lambda_2 = \)

The positive value of \(a\) for which the system of linear homogeneous equations \[ x+ay+z=0, \qquad ax+2y-z=0, \qquad 2x+3y+z=0 \] has non-trivial solution is

If the system of equations \[ 2x+3y-3z=3, \qquad x+2y+\alpha z=1, \qquad 2x-y+z=\beta \] has infinitely many solutions, then \[ \frac{\alpha}{\beta}-\frac{\beta}{\alpha} = \]