Matrices & Determinants Questions (2045)

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 $\sum_{r=1}^{n} \phi(r) = \begin{vmatrix} \sum_{r=1}^{n} f_1(r) & \sum_{r=1}^{n} f_2(r) \\ \sum_{r=1}^{n} f_3(r) & \sum_{r=1}^{n} f_4(r) \end{vmatrix}$

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 \(a, b, c, \lambda, m, n \in \mathbb{R} - \{0\}\) 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:

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

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:

Let $A = [a_{ij}]$ be a matrix of order $3 \times 3$, with $a_{ij} = (\sqrt{2})^{i+j}$. If the sum of all elements in the third row of $A^2$ is $\alpha + \beta\sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then $\alpha + \beta$ is equal to:

Let $S = \left\{m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6}\right\}$, where $A = \begin{bmatrix}2 & -1 \\ 1 & 0\end{bmatrix}$. Then $n(S)$ is equal to ___

Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3\,\text{adj}(-6\,\text{adj}(3A))) = 2^{m+n} \cdot 3^{mn}$, $m > n$. Then $4m + 2n$ is equal to ___

Let $\alpha, \beta$ ($\alpha \neq \beta$) be the values of $m$, for which the equations $x + y + z = 1$; $x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\displaystyle\sum_{n=1}^{10}(n^\alpha + n^\beta)$ is equal to:

If the system of equations $x+4y-z=\lambda$, $7x+9y+\mu z=-3$, $5x+y+2z=-1$ has infinitely many solutions, then $(2\mu+3\lambda)$ is equal to:

Consider the matrices $A=\begin{pmatrix}2&-5\\3&m\end{pmatrix}$, $B=\begin{pmatrix}20\\m\end{pmatrix}$ and $X=\begin{pmatrix}x\\y\end{pmatrix}$. Let the set of all $m$ for which $AX=B$ has a negative solution (i.e., $x<0$ and $y<0$) be the interval $(a,b)$. Then $8\int_a^b|A|\,dm$ is equal to ________.

Let $\lambda,\mu\in\mathbb{R}$. If the system of equations $3x+5y+\lambda z=3$, $7x+11y-9z=2$, $97x+155y-189z=\mu$ has infinitely many solutions, then $\mu+2\lambda$ is equal to:

Let $A$ be a non-singular matrix of order 3. If $\det(3\,\text{adj}(2\,\text{adj}((\det A)A)))=3^{-13}\cdot2^{-10}$ and $\det(3\,\text{adj}(2A))=2^m\cdot3^n$, then $|3m+2n|$ is equal to

Let $B=\begin{bmatrix}1&3\\1&5\end{bmatrix}$ and $A$ be a $2\times2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta-\alpha$ is equal to:

255. Let \(A = [a_{ij}]_{2\times 2}\) be a matrix where \(a_{ij} \in \{2, 3\}\). If determinant of matrix \(A\) is non-negative, then probability that it is invertible is:

Let \(A^T = A\) and \(B^T = B\). Consider the following statements:Statement-1: \(A(BA)^T = A(BA)\)Statement-2: \((AB)^T = B^T A^T = BA\) since \(AB\) is commutative.Which of the following is correct?

If a, b, c are sides of a scalene triangle, then the value of determinant \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\) is always:

61. If the system of equations\(x + y + z = 5\)\(x + 2y + 3z = 9\)\(x + 3y + \alpha z = \beta\)has infinitely many solutions, then \(\beta - \alpha\) equals ______.

921. If \(A\) and \(B\) are square matrices of order 3 such that \(2(A + B) = A^T + B^T + 3I\) and \(AA^T = 4I\), then find the value of \(\det.(12A^{-1} - BA^T + I)\).[Note: \(I\) is an identity matrix of order 3 and \(P^T\) denotes the transpose of matrix \(P\).]