Matrices and Determinants

If \({a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}\), r = 1, 2, 3, ....., i = \(\sqrt { - 1}\), then the determinant $$\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_4}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a...

If the following system of linear equations2x + y + z = 5x \(-\) y + z = 3x + y + az = bhas no solution, then :

The number of elements in the set \(\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}\), where I is 2 \(\times\) 2 identity matrix, is :

If \(\alpha\) + \(\beta\) + \(\gamma\) = 2\(\pi\), then the system of equations x + (cos \(\gamma\))y + (cos \(\beta\))z = 0(cos \(\gamma\))x + y + (cos \(\alpha\))z = 0(cos \(\beta\))x + (cos \(\alpha\))y + z = 0has :

Let \(f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|,x \in [0,\pi ]\). Then...

For real numbers \(\alpha\) and \(\beta\), consider the following system of linear equations :x + y \(-\) z = 2, x + 2y + \(\alpha\)z = 1, 2x \(-\) y + z = \(\beta\). If the system has infinite solutions, then \(\alpha\) + \(\beta\) is equa...

Let \(A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]\). If A\(-\)1 = \(\alpha\)I + \(\beta\)A, \(\alpha\), \(\beta\) \(\in\) R, I is a 2 \(\times\) 2 identity matrix then 4(\(\alpha\) \(-\) \(\beta\)) is equal to :

If \(A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]\) and M = A + A2 + A3 + ....... + A20, then the sum of all the elements of the matrix M is equal to _____________.

Let A and B be two 3 \(\times\) 3 real matrices such that (A2 \(-\) B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determinant of the matrix A3 + B3 is equal to :

If the system of linear equations2x + y \(-\) z = 3x \(-\) y \(-\) z = \(\alpha\)3x + 3y + \(\beta\)z = 3has infinitely many solution, then \(\alpha\) + \(\beta\) \(-\) \(\alpha\)\(\beta\) is equal to _____________.

If the matrix \(A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)\) satisfies \(A({A^3} + 3I) = 2I\), then the value of K is :

Let [\(\lambda\)] be the greatest integer less than or equal to \(\lambda\). The set of all values of \(\lambda\) for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [\(\lambda\)])z = [\(\lambda\)] has ...

Let A(a, 0), B(b, 2b + 1) and C(0, b), b \(\ne\) 0, |b| \(\ne\) 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is :

Let \(A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right)\), where [t] denotes the greatest integer less than or equal to t. If det(A) =...

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is :

The value of \(\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & 1 \cr {(a + 3)(a + 4)} & {a + 4} & 1 \cr } } \right|\) is :

If the matrix \(A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]\) satisfies the equation $${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 4 & 0 \cr
0 &...

Consider the following system of equations :x + 2y \(-\) 3z = a2x + 6y \(-\) 11z = bx \(-\) 2y + 7z = c,where a, b and c are real constants. Then the system of equations :

If \(A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)\), \(B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)\), $$i = \s...

Let \(\theta \in \left( {0,{\pi \over 2}} \right)\). If the system of linear equations\((1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0\)\({\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0\)$${\cos ^2...

Let A be a 3 \(\times\) 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right)\). Then A2025 \(-\) A2020 is equal to :

Let \(M = \left\{ {A = \left( {\matrix{ a & b \cr c & d \cr } } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}\). Define f : M \(\to\) Z, as f(A) = det(A), for all A\(\in\)M, where z is set of all integers. Then the ...

The values of a and b, for which the system of equations 2x + 3y + 6z = 8x + 2y + az = 53x + 5y + 9z = bhas no solution, are :

If \(P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]\), then P50 is :

The number of distinct real roots of \(\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0\) in the interval $$ - {\pi \over 4} \le x \l...

If \(A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr } } \right]\) and $$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{
a & { - b} \cr
b & a...

If the system of equationskx + y + 2z = 13x \(-\) y \(-\) 2z = 2\(-\)2x \(-\)2y \(-\)4z = 3has infinitely many solutions, then k is equal to __________.

Let \(A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]\), where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If \({A^2} = {I_3}\), then the value of \({x^3} + {y^3} + {z^3}\) is...

Let A be a 3 \(\times\) 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 \(\to\) 2R2 + 5R3 on 2A, then det(B) is equal to :

If for the matrix, \(A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]\), \(A{A^T} = {I_2}\), then the value of \({\alpha ^4} + {\beta ^4}\) is :

The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x \(-\) y + 4z = 8

Let P = \(\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]\), where \(\alpha\) \(\in\) R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k \(\in\) R. If ...

Let M be any 3 \(\times\) 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is ________.

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

For the system of linear equations:\(x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R\),consider the following statements :(A) The system has unique solution if \(k \ne 2,k \ne - 2\).(B) The system has unique solution if k = \(-\)2(C) The ...

Let A and B be 3 \(\times\) 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 \(-\) B2A2) X = O, where X is a 3 \(\times\) 1 column matrix of unknown variables and O is...

Let \(A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]\). Then the number of 3 \(\times\) 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________.

Let A = [aij] be a real matrix of order 3 \(\times\) 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :

The values of \(\lambda\) and \(\mu\) such that the system of equations \(x + y + z = 6\), \(3x + 5y + 5z = 26\), \(x + 2y + \lambda z = \mu\) has no solution, are :

Let a, b, c, d in arithmetic progression with common difference \(\lambda\). If \(\left| {\matrix{ {x + a - c} & {x + b} & {x + a} \cr {x - 1} & {x + c} & {x + b} \cr {x - b + d} & {x + d} & {x + c} \cr } } \right| = 2\), t...

Let \(A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)\) and B = 7A20 \(-\) 20A7 + 2I, where I is an identity matrix of order 3 \(\times\) 3. If B = [bij], then b13is equal to ___________...

Let \(A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]\), a\(\in\)R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of deter...

Let \(A = \{ {a_{ij}}\}\) be a 3 \(\times\) 3 matrix, where \({a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.\) then $$\...

The value of k \(\in\)R, for which the following system of linear equations3x \(-\) y + 4z = 3,x + 2y \(-\) 3z = \(-\)26x + 5y + kz = \(-\)3,has infinitely many solutions, is :

Consider the system of linear equations\(-\)x + y + 2z = 03x \(-\) ay + 5z = 12x \(-\) 2y \(-\) az = 7Let S1 be the set of all a\(\in\)R for which the system is inconsistent and S2 be the set of all a\(\in\)R for which the system has infini...

Let \(A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \cr } } \right]\) and \(2A - B = \left[ {\matrix{ 2 & { - 1} & 5 \cr 2 & { - 1} & 6 \cr 0 & 1 & 2 \cr } } \right]\). If Tr(A) ...

Let \(\alpha\), \(\beta\), \(\gamma\) be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c \(\in\) R and a, b \(\ne\) 0). If the system of equations (in u, v, w) given by \(\alpha\)u + \(\beta\)v + \(\gamma\)w = 0, \(\beta\)u ...

The solutions of the equation $$\left| {\matrix{
{1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr
{{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr
{4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr

} } \right| = 0...

Let the system of linear equations 4x + \(\lambda\)y + 2z = 02x \(-\) y + z = 0\(\mu\)x + 2y + 3z = 0, \(\lambda\), \(\mu\)\(\in\)R.has a non-trivial solution. Then which of the following is true?