Matrices and Determinants

If the system of linear equations
$$2x+y-z=7$$
$$x-3y+2z=1$$
$$x+4y+\delta z=k,\ \text{where }\delta,k\in\mathbf R$$
has infinitely many solutions, then $\delta+k$ is equal to :

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^2+A^3+\dots+A^{10}$ is equal to :

Let $A=\begin{pmatrix}2&-1\\0&2\end{pmatrix}$. If $B=I-{}^5C_1(\operatorname{adj}A)+{}^5C_2(\operatorname{adj}A)^2-\dots-{}^5C_5(\operatorname{adj}A)^5$, then the sum of all elements of the matrix B is

Let $M=\begin{bmatrix}0&-\alpha\\\alpha&0\end{bmatrix}$, where $\alpha$ is a non-zero real number an $N=\displaystyle\sum_{k=1}^{49}M^{2k}$. If $\left(I-M^2\right)N=-2I$, then the positive integral value of $\alpha$ is ______.

Let $A=\begin{bmatrix}1&-2&\alpha\\\alpha&2&-1\end{bmatrix}$ and $B=\begin{bmatrix}2&\alpha\\-1&2\\4&-5\end{bmatrix}$, $\alpha\in\mathbf C$. Then the absolute value of the sum of all values of $\alpha$ for which $\det(AB)=0$ is :

Let A and B be two square matrices of order 2. If $\det(A)=2$, $\det(B)=3$ and $\det\left(\left(\det\left(5(\det A)B\right)\right)A^2\right)=2^a3^b5^c$ for some $a,b,c\in\mathbf N$, then $a+b+c$ is equal to :

Suppose $\displaystyle\lim_{x\to0}\frac{F(x)}{x^3}$ exists and is equal to L, where
$$F(x)=\begin{vmatrix}a+\sin\frac x2&-b\cos x&0\\-b\cos x&0&a+\sin\frac x2\\0&a+\sin\frac x2&-b\cos x\end{vmatrix}.$$
Then, $-112L$ is equal to ______.

The number of $\theta\in(0,4\pi)$ for which the system of linear equations
$$3(\sin3\theta)x-y+z=2$$
$$3(\cos2\theta)x+4y+3z=3$$
$$6x+7y+7z=9$$
has no solution, is :

Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$. If $\omega=\dfrac{\sqrt3i-1}2$, then the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n+(\omega B)^n=A+B\right\}$ is equal to ______.

The number of real values of $\lambda$, such that the system of linear equations
$$2x-3y+5z=9$$
$$x+3y-z=-18$$
$$3x-y+\left(\lambda^2-|\lambda|\right)z=16$$
has no solutions, is

Let $A=\begin{bmatrix}1&a&a\\0&1&b\\0&0&1\end{bmatrix}$, $a,b\in\mathbb R$. If for some $n\in\mathbb N$, $A^n=\begin{bmatrix}1&48&2160\\0&1&96\\0&0&1\end{bmatrix}$ then $n+a+b$ is equal to ______.

If the system of linear equations.
$$8x+y+4z=-2$$
$$x+y+z=0$$
$$\lambda x-3y=\mu$$
has infinitely many solutions, then the distance of the point $\left(\lambda,\mu,-\dfrac12\right)$ from the plane $8x+y+4z+2=0$ is :

Let A be a $2\times2$ matrix with $\det(A)=-1$ and $\det((A+I)(\operatorname{Adj}(A)+I))=4$. Then the sum of the diagonal elements of A can be :

Let $A=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $B=\begin{bmatrix}9^2&-10^2&11^2\\12^2&13^2&-14^2\\-15^2&16^2&17^2\end{bmatrix}$, then the value of $A'BA$ is:

The number of matrices $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d\in\{-1,0,1,2,3,\ldots,10\}$, such that $A=A^{-1}$, is ______.

Let $A=\begin{pmatrix}1&2\\-2&-5\end{pmatrix}$. Let $\alpha,\beta\in\mathbf R$ be such that $\alpha A^2+\beta A=2I$. Then $\alpha+\beta$ is equal to

Let S be the set containing all $3\times3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{\mathrm T}A$ is 6 is ______.

Let $A=\begin{pmatrix}4&-2\\\alpha&\beta\end{pmatrix}$.

If $A^2+\gamma A+18I=O$, then $\det(A)$ is equal to ______.

Consider a matrix $A=\begin{bmatrix}\alpha&\beta&\gamma\\\alpha^2&\beta^2&\gamma^2\\\beta+\gamma&\gamma+\alpha&\alpha+\beta\end{bmatrix}$, where $\alpha,\beta,\gamma$ are three distinct natural numbers.

If $\dfrac{\det(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32}\times3^{16}$, then the number of such 3-tuples $(\alpha,\beta,\gamma)$ is ______.

Let the vectors $\vec a=(1+t)\hat i+(1-t)\hat j+\hat k$, $\vec b=(1-t)\hat i+(1+t)\hat j+2\hat k$ and $\vec c=t\hat i-t\hat j+\hat k$, $t\in\mathbf R$ be such that for $\alpha,\beta,\gamma\in\mathbf R$, $\alpha\vec a+\beta\vec b+\gamma\vec c=\vec0\Rightarrow\alpha=\beta=\gamma=0$. Then, the set of all values of t is :