Matrices and Determinants

For the system of linear equations
\(2x - y + 3z = 5\)
\(3x + 2y - z = 7\)
\(4x + 5y + \alpha z = \beta\),
which of the following is NOT correct?

If A is a 3 \(\times\) 3 matrix and \(|A| = 2\), then \(|3\,adj\,(|3A|{A^2})|\) is equal to :

Let \(\mathrm{S}\) be the set of values of \(\lambda\), for which the system of equations \(6 \lambda x-3 y+3 z=4 \lambda^{2}\), \(2 x+6 \lambda y+4 z=1\), \(3 x+2 y+3 \lambda z=\lambda\) has no solution. Then $$12 \sum_\limits{i \in S}|\la...

If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$, then \(|\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))|\) is equal to :

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

Let \(A = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]\) and \(B = \left[ {\matrix{ 2 & \alpha \cr { - 1} & 2 \cr 4 & { - 5} \cr } } \right],\,\alpha \in C\). Then the absolut...

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 A2 + A3 + ...... + A10 is equal to :

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

Let \(M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]\), where \(\alpha\) is a non-zero real number an \(N = \sum\limits_{k = 1}^{49} {{M^{2k}}}\). If \((I - {M^2})N = - 2I\), then the positive integral v...

Let \(A = \left( {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right)\). If \(B = I - {}^5{C_1}(adj\,A) + {}^5{C_2}{(adj\,A)^2} - \,\,.....\,\, - {}^5{C_5}{(adj\,A)^5}\), then the sum of all elements of the matrix B is

Let p and p + 2 be prime numbers and let
$$
\Delta=\left|\begin{array}{ccc}
\mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\
(\mathrm{p}+1) ! & (\mathrm{p}+2) ! & (\mathrm{p}+3) ! \\
(\mathrm{p}+2) ! & (\mathrm{p}+3) ! & (\mathrm{p}+4)...

Let A and B be two \(3 \times 3\) non-zero real matrices such that AB is a zero matrix. Then

Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$$. For \(\mathrm{k} \in N\), if \(X^{\prime} A^{k} X=33\), then \(\mathrm{k}\) is equal to...

If the system of equations
$$ \begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned} $$
has infinitely many solutions, then \(\alpha+\beta\) is equal to

Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementary row operation ?

Let A be a matrix of order 3 \(\times\) 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________.

If the system of linear equations
\(2x + 3y - z = - 2\)
\(x + y + z = 4\)
\(x - y + |\lambda |z = 4\lambda - 4\)
where, \(\lambda\) \(\in\) R, has no solution, then

Let \(A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)\) where \(i = \sqrt { - 1}\). Then, the number of elements in the set { n \(\in\) {1, 2, ......, 100} : An = A } is ____________.

If the system of linear equations \(2x - 3y = \gamma + 5\), \(\alpha x + 5y = \beta + 1\), where \(\alpha\), \(\beta\), \(\gamma\) \(\in\) R has infinitely many solutions then the value of | 9\(\alpha\) + 3\(\beta\) + 5\(\gamma\) | is equ...

Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \beta \in \mathbf{R}$$. Let \(\alpha_{1}\) be the value of \(\alpha\) which satisfies $$(\m...

Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix \(B_{0}=A^{49}+2 A^{98}\). If \(B_{n}=A d j\left(B_{n-1}\right)\) for all \(n \geq 1\), then $$\operatorname{det}\left(B_{4}\r...

Let \(\mathrm{A}\) and \(\mathrm{B}\) be any two \(3 \times 3\) symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

The positive value of the determinant of the matrix A, whose
Adj(Adj(A)) = \(\left( {\matrix{ {14} & {28} & { - 14} \cr { - 14} & {14} & {28} \cr {28} & { - 14} & {14} \cr } } \right)\), is _____________.

Let the system of linear equations \(x + 2y + z = 2\), \(\alpha x + 3y - z = \alpha\), \(- \alpha x + y + 2z = - \alpha\) be inconsistent. Then \(\alpha\) is equal to :

Let A and B be two 3 \(\times\) 3 matrices such that \(AB = I\) and \(|A| = {1 \over 8}\). Then \(|adj\,(B\,adj(2A))|\) is equal to

Let \(f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R\). Then the sum of the squares of all the values of a, for which \(2f'(10) - f'(5) + 100 = 0\), is

Let \(S\) be the set containing all \(3 \times 3\) 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 $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha A^{2}+\beta A=2 I\). Then \(\alpha+\beta\) is equal to

Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$$, where \(\alpha, \beta, \gamma\) are three distinct natural numb...

Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$.
If \(\mathrm{A}^{2}+\gamma \mathrm{A}+18 \mathrm{I}=\mathrm{O}\), then \(\operatorname{det}(\mathrm{A})\) is equal to _____________.

The ordered pair (a, b), for which the system of linear equations
3x \(-\) 2y + z = b
5x \(-\) 8y + 9z = 3
2x + y + az = \(-\)1
has no solution, is :

Let A be a 3 \(\times\) 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :

Let \(X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta X + \gamma {X^2}\) and \(Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}\), \(\alpha\), $$\beta$...

If the system of equations
\(\alpha\)x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = \(\beta\)
has infinitely many solutions, then the ordered pair (\(\alpha\), \(\beta\)) is equal to :

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

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 \(\left( {\lambda ,\mu , - {1 \over 2}} \right)\) from the plane $$8x + y + ...

The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where \(a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}\), such that \(A=A^{-1}\), is ___________.

$$
\text { Let } A=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
9^{2} & -10^{2} & 11^{2} \\
12^{2} & 13^{2} & -14^{2} \\
-15^{2} & 16^{2} & 17^{2}
\end{array}\right] \text {, then the value ...

Let \(A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]\). If M and N are two matrices given by \(M = \sum\limits_{k = 1}^{10} {{A^{2k}}}\) and \(N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}}\) then MN2 is :

Let A be a 3 \(\times\) 3 real matrix such that
$$A\left( {\matrix{
1 \cr
1 \cr
0 \cr

} } \right) = \left( {\matrix{
1 \cr
1 \cr
0 \cr

} } \right);A\left( {\matrix{
1 \cr
0 \cr
1 \cr

} } \r...

Let \(A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)\) and \(B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)\). Then the number of elements in the set {(n, m) : n, m \(\in\) {1, 2, .......

The system of equations
\(- kx + 3y - 14z = 25\)
\(- 15x + 4y - kz = 3\)
\(- 4x + y + 3z = 4\)
is consistent for all k in the set

Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and \(B=A-I\). If \(\omega=\frac{\sqrt{3} i-1}{2}\), then the number of elements in the $$\operatorname{set}\left\{n \in\{1,2, \ldots, 100\}: A^{n}+...

The number of \(\theta \in(0,4 \pi)\) for which the system of linear equations

$$ \begin{aligned} &3(\sin 3 \theta) x-y+z=2 \\\\ &3(\cos 2 \theta) x+4 y+3 z=3 \\\\ &6 x+7 y+7 z=9 \end{aligned} $$
has no solution, is :

Let $$A=\left[\begin{array}{lll} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right], a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}, A^{n}=\left[\begin{array}{ccc} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{array}\right] $$ th...

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

Let S = {\(\sqrt{n}\) : 1 \(\le\) n \(\le\) 50 and n is odd}.
Let a \(\in\) S and \(A = \left[ {\matrix{ 1 & 0 & a \cr { - 1} & 1 & 0 \cr { - a} & 0 & 1 \cr } } \right]\).
If $$\sum\limits_{a\, \in \,S}^{} {\det (adj\,A) = ...

The number of values of \(\alpha\) for which the system of equations :
x + y + z = \(\alpha\)
\(\alpha\)x + 2\(\alpha\)y + 3z = \(-\)1
x + 3\(\alpha\)y + 5z = 4
is inconsistent, is

Let \(S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right\}\) and let \({T_n} = \{ A \in S:{A^{n(n + 1)}} = I\}\). Then the number of elements in $$\bigcap\limits_{n = 1}^{100}...

Let the system of linear equations
x + y + \(\alpha\)z = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x\(^ *\), y\(^ *\), z\(^ *\)). If (\(\alpha\), x\(^ *\)), (y\(^ *\), \(\alpha\)) and (x\(^ *\), \(-\)y\(^ *\)) are collinear...