IISER Mathematics - Determinants and Matrices
Exam Duration: 45 Mins Total Questions : 30
The value of the determinant, where \(a\neq b\neq c\),
\(\left| \begin{matrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{matrix} \right| \) is
- (a)
-1
- (b)
2
- (c)
1
- (d)
0
If a,b,c are all positive and pth,qth,rth terms respectively of a G.P.,then the value of the determinant is
\(\left| \begin{matrix} log\quad a & p & 1 \\ log\quad b & q & 1 \\ log\quad c & r & 1 \end{matrix} \right| \) is
- (a)
pqr
- (b)
p+q+r-abc
- (c)
0
- (d)
npne of these
If, \(A=\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), then the matrix A2-9A+14I2, equals
- (a)
A
- (b)
-9A
- (c)
I2
- (d)
0
If \(R(t)=\left[ \begin{matrix} cos\quad t & sin\quad t \\ -sin\quad t & cos\quad t \end{matrix} \right] ;\) then R(t) R(s) equals
- (a)
R(s-t)
- (b)
R(s+t)
- (c)
R(t-s)
- (d)
none of these
The inverse of diagonal matrix is
- (a)
symmetric matrix
- (b)
skew-symmetric matrix
- (c)
diagonal matrix
- (d)
none of these
In a system of linear equations, if A is the coefficient matrix such that \(|A|\)=0, then the system of equations
- (a)
has a unique solution
- (b)
has no solution
- (c)
has an infinite number of solutions
- (d)
is either inconsistent or has an infinite number of solutions
If \(A=\left[ \begin{matrix} a & b \\ b & a \end{matrix} \right] and\quad { A }^{ 2 }=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \end{matrix} \right] \), then
- (a)
\(\alpha ={ a }^{ 2 }+{ b }^{ 2 },\beta =ab\)
- (b)
\(\alpha ={ a }^{ 2 }+{ b }^{ 2 },\beta =2ab\)
- (c)
\(\alpha =2ab,\beta ={ a }^{ 2 }+{ b }^{ 2 }\)
- (d)
\(\alpha ={ a }^{ 2 }+{ b }^{ 2 },\beta ={ a }^{ 2 }-{ b }^{ 2 }\)
What is the value of (A+I)3+(A+I)3-6A, when A3=I (A=square matrix)?
- (a)
3I
- (b)
2I
- (c)
4A
- (d)
3A
If a and B are square matrices of size nXn such that A2-B2=(A-B)(A+B), then which of the following will be always true?
- (a)
AB=BA
- (b)
Either of A or B is a zero matrix
- (c)
Either of A or B is an identity matrix
- (d)
A=B
If x=-4 is a root of \(\begin{vmatrix} x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x \end{vmatrix}=0\) then possible roots are
- (a)
x=-4,1,3
- (b)
x=4,1,3
- (c)
x=3,-1,4
- (d)
x=-4,-1,3
Use product \(\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}\) for equations x-y+2z=1, 2y-3z=1, 3x-2y+4z=2 then values of x,y and z will be equal to
- (a)
x=0,y=5, z=3
- (b)
x=1, y=5, z=2
- (c)
x=0,y=5,z=0
- (d)
x=0, y=5,z=2
If a,b,c are the sides of a \(\Delta ABC\)opposite to angles A,B,C respectively, then \(\Delta =\begin{vmatrix} a2 & bsinA & csinA \\ b\quad sinA & 1 & cos(B-C) \\ c\quad sinA & cos(B-) & 1 \end{vmatrix}\) is equal to
- (a)
sin A - sin C sin B
- (b)
abc
- (c)
1
- (d)
0
If P=\(\begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}\) is the adjoint of a 3X3 matrix A and \(\begin{vmatrix} A \end{vmatrix}=4\) then \(\alpha\) is equal to
- (a)
4
- (b)
11
- (c)
5
- (d)
0
If \(D=\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}\) for \(x\ne0\), \(y\ne0\) then D is
- (a)
Divisible by neither x nor y
- (b)
divisible by both x and y
- (c)
divisible by x but not y
- (d)
divisible by y but not x
Let A and B are two matrices of same order 3 x 3, where A = \(\left( \begin{matrix} 1 & 3 & \lambda +2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{matrix} \right) \), B = \(\left( \begin{matrix} 3 & 2 & 4 \\ 3 & 2 & 5 \\ 2 & 1 & 4 \end{matrix} \right) \)
If \(\lambda\).= 3, then \(\frac{1}{7}\)(tr(AB) + tr(BA)) is equal to
- (a)
34
- (b)
42
- (c)
84
- (d)
63
If A5 = 0 such that An \(\neq \) I for \(1\le n\le 4\) then (I - A)-1 is equal to
- (a)
A4
- (b)
A3
- (c)
I + A
- (d)
none of these
Let \(\Delta \left( x \right) =\left| \begin{matrix} x+a & x+b & x+a-c \\ x+b & x+c & x-1 \\ x+c & x+d & x-b+d \end{matrix} \right| \) and \(\int _{ 0 }^{ 2 }{ \Delta \left( x \right) dx } =-16\) where a, b, c, d are in AP, then the common difference of the AP is equal to
- (a)
\(\pm 1\)
- (b)
\(\pm 2\)
- (c)
\(\pm 3\)
- (d)
\(\pm4\)
The determinant \(\Delta =\left| \begin{matrix} { a }^{ 2 }+{ x }^{ 2 } & { a }^{ 2 } & { a }^{ 2 } \\ { b }^{ 2 } & { b }^{ 2 }+{ x }^{ 2 } & { b }^{ 2 } \\ { c }^{ 2 } & { c }^{ 2 } & { c }^{ 2 }+{ x }^{ 2 } \end{matrix} \right| \) is divisible by
- (a)
x
- (b)
x2
- (c)
x3
- (d)
x4
If P =\(\left[ \begin{matrix} i & 0 & -i \\ 0 & -i & i \\ -i & i & 0 \end{matrix} \right] \) and Q =\(\left[ \begin{matrix} -i & i \\ 0 & 0 \\ i & -i \end{matrix} \right] \) , then PQ is equal to
- (a)
\(\left[ \begin{matrix} -2 & 2 \\ 1 & -1 \\ 1 & -1 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 2 & -2 \\ -1 & 1 \\ -1 & 1 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 2 & -2 \\ -1 & 1 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \)
Let A =\(\left[ \begin{matrix} cos^{ 2 }\theta & sin\theta cos\theta \\ cos\theta sin\theta & sin^{ 2 }\theta \end{matrix} \right] \)and B=\(\left[ \begin{matrix} cos^{ 2 }\phi & sin\phi cos\phi \\ cos\phi sin\phi & sin^{ 2 }\phi \end{matrix} \right] \) , then AB =O , if
- (a)
\(\theta =n\phi ,n=0,1,2...\)
- (b)
\(\theta +\phi =n\pi =0,1,2...\)
- (c)
\(\theta =\phi +(2n+1)\frac { \pi }{ 2 } =0,1,2...\)
- (d)
\(\theta =\phi +\frac { n\pi }{ 2 } n=0,1,2...\)
Evaluate the following determinant \(\left| \begin{matrix} cos\ \theta & -sin\ \theta \\ sin\ \theta & cos\ \theta \end{matrix} \right| \)
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Find the values of x,y,z respectively if the matrix A =\(\left[ \begin{matrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{matrix} \right] \) satisfy the equation ATA =I3
- (a)
\(\frac { 1 }{ \sqrt { 2 } } ,\frac { 1 }{ \sqrt { 6 } } ,\frac { 1 }{ \sqrt { 3 } } \)
- (b)
\(\frac { -1 }{ \sqrt { 2 } } ,\frac { -1 }{ \sqrt { 6 } } ,\frac { -1 }{ \sqrt { 3 } } \)
- (c)
Both (a) & (b)
- (d)
None of these
Find the adjoint of the matrix A=\(\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right] \)
- (a)
\(\left[ \begin{matrix} 4 & 2 \\ 3 & 1 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 4 & -2 \\ -3 & 1 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 1 & -2 \\ -3 & 4 \end{matrix} \right] \)
For any square matrix A, AAT is a
- (a)
unit matrix
- (b)
symmetric matrix
- (c)
skew-symmetric matrix
- (d)
diagonal matrix
If ω is a complex cube root of unity, then the matrix A=\(\left[ \begin{matrix} 1 & { \omega }^{ 1 } & { \omega } \\ { \omega }^{ 2 } & { \omega } & 1 \\ { \omega } & 1 & { { \omega } }^{ 2 } \end{matrix} \right] \) is
- (a)
symmetric matrix
- (b)
diagonal matrix
- (c)
skew-symmetric matrix
- (d)
none of these
If A =\(\left[ \begin{matrix} 4 & 3 & -1 \\ 3 & 5 & 7 \\ 1 & -2 & 1 \end{matrix} \right] \) is the sum of a symmetric matrix B and a skew-symmetric matrix C, then B is
- (a)
\(\left[ \begin{matrix} 1 & 5/2 & 0 \\ 5/2 & 5 & 5/2 \\ 1 & 5/2 & 1 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 4 & 5/2 & 0 \\ 5/2 & 5 & 5/2 \\ 0 & 5/2 & 1 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 1 & 5/2 & 0 \\ 5/2 & 5 & 5/2 \\ 0 & 5/2 & 2 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 1 & 5/2 & 0 \\ 5/2 & 5 & 3/2 \\ 0 & 5/2 & 2 \end{matrix} \right] \)
Find the value of the following determinants. \(\left| \begin{matrix} { (b+c) }^{ 2 } & { a }^{ 2 } & { bc } \\ { (c+a) }^{ 2 } & { b }^{ 2 } & ca \\ { (a+b) }^{ 2 } & { c }^{ 2 } & ab \end{matrix} \right| =\)
- (a)
(a-b)(b-c)(c-a)(a2+b2+c2)
- (b)
-(a-b)(b-c)(c-a)
- (c)
(a-b)(b-c)(c-a)(a+b+c)(a2+b2+c2)
- (d)
0
If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k.
- (a)
4
- (b)
7/140
- (c)
47
- (d)
40/7
If 4x + 3y + 6z = 25, x+5y + 7z = 13, 2x + 9y + z = 1, then z =
- (a)
1
- (b)
3
- (c)
-2
- (d)
2
A diet is to contain 30 units of vitamin A, 40 units of vitamin Band 20 units of vitamin C. Three types of foods F1, F2 and F3 are available. 1 unit offood F1 contains 3 units of vitamin A, 2 units of vitamin B, 1 unit of vitamin C. 1 unit of food F2 contains 1 unit of vitamin A, 2 units of vitamin Band 1 unit of vitamin C. 1unit of food F3 contains 5 units of vitamin A, 3 units of vitamin Band 2 units of vitamin C. Represent the above situation algebraically and find the diet contains each types of food by using matrix method.
- (a)
2, 15, 0
- (b)
5, 15, 10
- (c)
5, -15, 0
- (d)
-5, 15, 0