Determinants Questions (2072)

Let α be a root of the equation x^2 + x + 1 = 0 and the matrix A = 1/sqrt(3) * [[1, 1, 1], [1, α, α^2], [1, α^2, α^4]], then the matrix A^31 is equal to:

Which of the following values of α satisfy the equation ?

Let a - 2b + c = 1. If f(x) = x+ax+2x+1x+bx+3x+2x+cx+4x+3, then :

If system of equation a1x + b1y = c1 & a2x + b2y = c2 (where a1, b1, c1, a2, b2, c2 ≠ 0) has infinite solutions, then-(A) a1/a2 = b1/b2 = c1/c2(B) (a1+a2)/(a1-a2) = (b1+b2)/(b1-b2) = (c1+c2)/(c1-c2)(C) the quadratic equations a1x2 + b1x + c1 = 0 & a2x2 + b2x + c2 = 0 have no common root(D) system of equation a12 a2x + b12 b2y = c12 c2 & a1 a22x + b1 b22y = c1 c22 will also have infinite number of solutions

16. Consider the row sums \(R_i = \sum_{j=1}^{n} a_{ij}\) (\(i = 1, 2, \ldots, n\)) and the column sums \(C_j = \sum_{i=1}^{n} a_{ij}\) (\(j = 1, 2, \ldots, n\)). Let \(p\) be the smallest of all these sums \(R_i\) and \(C_j\), i.e., \(p = \min_{i,j}\{R_i, C_j\}\). Show that \(S > n^2/2\), where \(S\) is the sum of all elements of the matrix. What is the value of \(p\) (as a fraction of \(n^2/2\)) in the minimum case?

Let $A = \begin{bmatrix} \frac{1}{2} & -\frac{3}{2} \\ 1 & -\frac{1}{2} \end{bmatrix}$, then the value of sum of all the elements of $A^{100}$ is

NTA Test 22 (Single Choice)

If \(A\) and \(B\) are two matrices of order \(3 \times 3\) where \(|A| = -2\), \(|B| = 2\), then \(|(A^{-1}\text{adj}(B^{-1})\text{adj}(2A^{-1})|\) is equal to

Let $M$ and $N$ be square matrices of the same order satisfying $MN = M$ and $NM = N$. Then $(M^{2024} + N^{2024})^{2025}$ is equal to

Let \(D = \begin{bmatrix}1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\end{bmatrix}\) and \(P = \begin{bmatrix}7 & 0 & 2\\ 0 & 1 & 0\\ 2 & 0 & 5\end{bmatrix}\). Consider \(A = P^{-1}DP\). Find \(\det.(A^2 + A)\).

If the system of equations $2x+7y+\lambda z=3$, $3x+2y+5z=4$, $x+\mu y+32z=-1$ has infinitely many solutions, then $\lambda-\mu$ is equal to ________.

If determinant has two identical columns, then value is:

If two rows are proportional, determinant equals:

Let \(a, b, c \in \mathbb{R}\), not all equal, and \[\Delta_1 = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\], \[\Delta_2 = \begin{vmatrix} a+2b & b+3c & c+4a \\ b+2c & c+3a & a+4b \\ c+2a & a+3b & b+4c \end{vmatrix}\] then \(\dfrac{\Delta_2}{\Delta_1} =\) ________.

Let \(D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}\) and \(P = \begin{bmatrix} 7 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 5 \end{bmatrix}\). Consider \(A = P^{-1}DP\). Find \(\det.(A^2 + A)\).

Number of values of \(\theta\) lying in \([0, 100\pi]\) for which the system of equations \[(\cos 3\theta)x - y + z = 0\] \[(\cos 2\theta)x + 4y + 3z = 0\] \[2x + 7y + 7z = 0\] has non-trivial solution is ________.

Let \[D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}\] and \[D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}\] then the value of \(\dfrac{D_1}{D_2}\), where \(b \neq 0\) and \(ad \neq bc\), is ________.

761. Find the values of \(\mu\), \(\lambda\), and \(\gamma\) such that the system of equations is consistent. Given answer: \(\mu = 3\), \(\lambda = 12\), \(\gamma = 2\). Find \(\mu + \lambda + \gamma\).

Question 82: If $\phi(r) = \begin{vmatrix} r & r-1 \\ r-3 & r-4 \end{vmatrix}$, then $\sum_{r=1}^{n} \phi(r) = ?$Statement-1 (Assertion): $\sum_{r=1}^{n} \phi(r) = \frac{n(n+1)}{2}$Statement-2 (Reason): If $\phi(r) = \begin{vmatrix} f_1(r) & f_2(r) \\ f_3(r) & f_4(r) \end{vmatrix}$, then $\sum_{r=1}^{n} \phi(r) = \begin{vmatrix} \sum_{r=1}^{n} f_1(r) & \sum_{r=1}^{n} f_2(r) \\ \sum_{r=1}^{n} f_3(r) & \sum_{r=1}^{n} f_4(r) \end{vmatrix}$

Show that $\begin{vmatrix} a^2 + \lambda^2 & ab + c\lambda & ca - b\lambda \\ ab - c\lambda & b^2 + \lambda^2 & bc + a\lambda \\ ac + b\lambda & bc - a\lambda & c^2 + \lambda^2 \end{vmatrix} \times \begin{vmatrix} \lambda & c & -b \\ -c & \lambda & a \\ b & -a & \lambda \end{vmatrix} = \lambda^3(\lambda^2 + a^2 + b^2 + c^2)^3$

The values of \lambda and m for which the system of linear equations x + y + z = 2, x + 2y + 3z = 5, x + 3y + \lambda z = m has infinitely many solutions are, respectively

Let \[A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\] If \(A^n = m^{n-1} A\), and \(A^{18} = 16^{17} A = 2^{68} A\), find the number of factors of 68.

If the system of equations $x + \lambda y + 1 = 0, \lambda x + y + 1 = 0$ & $x + y + \lambda = 0$ is consistent, then find the value of $\lambda$.

If \(a, b, c, \lambda, m, n \in \mathbb{R} - \{0\}\) such that \(a\lambda + bm + cn = 0, b\lambda + cm + an = 0, c\lambda + am + bn = 0\). If \(a, b, c\) are distinct and \(f(x) = ax^3 + bx^2 + cx + 2\). Find \(f(1)\):

If \(\Delta\) = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and \(\Delta\) = 0, then rows are:

If $\begin{vmatrix}\sin x+1&\sin 2x&\sin 3x\\ \sin 2x&\sin 3x+a&\sin 4x\\ \sin 3x&\sin 4x&\sin 5x+a^2\end{vmatrix}=2025(f(x)+45)$ where $f(x)$ is a function of $x$ and $a$ is complex, then sum of all possible values of $a$ is

If the system of equations $x+y+az=b$, $2x+5y+2z=6$, $x+2y+3z=3$ has infinitely many solutions, then $2a+3b$ is equal to

Let $D_k=\begin{vmatrix}1&2k&2k-1\\n&n^2+n+2&n^2\\n&n^2+n&n^2+n+2\end{vmatrix}$. If $\displaystyle\sum_{k=1}^n D_k=96$, then $n$ is equal to _________.

If the system $7x+11y+\alpha z=13$, $5x+4y+7z=\beta$, $175x+194y+57z=361$ has infinitely many solutions, then $\alpha+\beta+2$ is equal to

Let $P=\begin{pmatrix}\frac{\sqrt3}{2}&\frac12\\-\frac12&\frac{\sqrt3}{2}\end{pmatrix}$, $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and $Q=PAP^T$. If $P^TQ^{2007}P=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, then $2a+b-3c-4d$ is equal to

Let $A=\begin{pmatrix}0&1&2\\a&0&3\\1&c&0\end{pmatrix}$, $a,c\in\mathbb{R}$. If $A^3=A$ and the positive value of $a$ belongs to $(n-1,n]$, $n\in\mathbb{N}$, then $n$ is equal to ____.

Let $B=\begin{pmatrix}1&3&\alpha\\1&2&3\\\alpha&\alpha&4\end{pmatrix}$, $\alpha>2$, be the adjoint of matrix $A$ with $|A|=2$. Then $[\alpha\ -2\alpha\ \alpha]B\begin{pmatrix}\alpha\\-2\alpha\\\alpha\end{pmatrix}$ is equal to

The number of symmetric matrices of order 3, with all entries from $\{0,1,2,3,4,5,6,7,8,9\}$ is

Let $\det(A)=m-n$ where $4m+n=22$ and $17m+4n=93$. If $\det(n\,\text{adj}(\text{adj}(mA)))=3^a5^b6^c$, then $a+b+c$ is equal to

507. If \(a^2 + 8b^2 + 2c^2 + 2d^2 - 4ab - 4bc - 4bd = 0\) (where \(a, b, c, d \in \mathbb{R}\)), then the value of \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is:

If the system $x+2y-3z=2,\ 2x+\lambda y+5z=5,\ 14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to:

If $f(x)=\begin{vmatrix}2\cos^4 x&2\sin^4 x&3+\sin^2 2x\\3+2\cos^4 x&2\sin^4 x&\sin^2 2x\\2\cos^4 x&3+2\sin^4 x&\sin^2 2x\end{vmatrix}$, then $\frac{1}{5}f'(0)$ is equal to

Let $A=\begin{bmatrix}1&0&0\\0&\alpha&\beta\\0&\beta&\alpha\end{bmatrix}$ and $|2A|^3=2^{21}$ where $\alpha,\beta\in\mathbb{Z}$. Then a value of $\alpha$ is

For $\alpha,\beta\in\mathbb{R}$ and a natural number $n$, let $A_r=\begin{vmatrix}r & 1 & \frac{n^2}{2}+\alpha\\ 2r & 2 & n^2-\beta\\ 3r-2 & 3 & \frac{n(3n-1)}{2}\end{vmatrix}$. Then $\displaystyle\sum_r A_r$ is:

| sin x cos x sin$x + cos$$x + 1$| 2 | | d y If y(x) = | 27 28 27 | ,x$\ in $R , then$2 + y$is equal to dx | 1 1 1 |

If the system of linear equations$3x + y$+$\betaz$= 3 2x +$\alphay$- z = -3$x + 2y + z = 4$has infinitely many solutions, then the value of 22$\beta$- 9$\alpha$is :

Let $\alpha\in(0,\infty)$ and $A=\begin{bmatrix}1&2&\alpha\\1&0&1\\0&1&2\end{bmatrix}$. If $\det(\text{adj}(2A-A^T)\cdot\text{adj}(A-2A^T))=2^8$, then $(\det(A))^2$ is equal to:

Let $A$ be a $3\times3$ matrix of non-negative real elements such that $A\begin{bmatrix}1\\1\\1\end{bmatrix}=3\begin{bmatrix}1\\1\\1\end{bmatrix}$. Then the maximum value of $\det(A)$ is

Let the system of equations: $$2x + 3y + 5z = 9$$ $$7x + 3y - 2z = 8$$ $$12x + 3y - (4 + \lambda)z = 16 - \mu$$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4x = 3y$ is

If the system of equations $x+2ay+az=0$, $x+3by+bz=0$, $x+4cy+cz=0$ has a non-zero solution, then $a,b,c$

Let $A=[a_{ij}]=\begin{pmatrix}\log_{5}128 & \log_{4}5\\ \log_{5}8 & \log_{4}25\end{pmatrix}$. If $A_{ij}$ is the cofactor of $a_{ij}$, and $C_{ij}=\displaystyle\sum_{k=1}^{2}a_{ik}A_{jk},\ 1\le i,j\le 2,\,C=[C_{ij}]$, then $8|C|$ is equal to:

Let $A$ be a $3\times 3$ matrix such that $X^{T}AX=O$ for all nonzero $3\times 1$ matrices $X=\begin{pmatrix}x\\y\\z\end{pmatrix}$. If $A\!\begin{pmatrix}1\\1\\1\end{pmatrix}=\begin{pmatrix}1\\4\\-5\end{pmatrix}$ and $A\!\begin{pmatrix}1\\2\\1\end{pmatrix}=\begin{pmatrix}0\\4\\-8\end{pmatrix}$, then $\det\bigl(\operatorname{adj}(2(A+I))\bigr)=2^{\alpha}\cdot 3^{\beta}\cdot 5^{\gamma}$ and $\alpha^{2}+\beta^{2}+\gamma^{2}$ is \rule{2cm}{0.4pt}.

If determinant is identity matrix, value is:

Let $A$ be a $3\times3$ matrix and $\det(A)=2$. If $n=\det(\underbrace{\text{adj}(\text{adj}(\cdots(\text{adj}\,A)\cdots))}_{\text{2024 times}})$, then the remainder when $n$ is divided by $9$ is equal to

If the system of equations $(\lambda-1)x + (\lambda-4)y + \lambda z = 5$, $\lambda x + (\lambda-1)y + (\lambda-4)z = 7$, $(\lambda+1)x + (\lambda+2)y - (\lambda+2)z = 9$ has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to: