Linear Algebra · Discrete Mathematics · GATE CSE
Marks 1
If $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$, then which ONE of the following is $A^8$ ?
Let $L, M$, and $N$ be non-singular matrices of order 3 satisfying the equations $L^2=L^{-1}, M=L^8$ and $N=L^2$. Which ONE of the following is the value of the determinant of $(M-N)$ ?
Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a realvalued constant. Which of the following option(s) is/are CORRECT?
$$\begin{aligned} & x+k y=1 \\ & k x+y=-1 \end{aligned}$$
The product of all eigenvalues of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is
The Lucas sequence $$L_n$$ is defined by the recurrence relation:
$${L_n} = {L_{n - 1}} + {L_{n - 2}}$$, for $$n \ge 3$$,
with $${L_1} = 1$$ and $${L_2} = 3$$.
Which one of the options given is TRUE?
Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 & 2 & 3 & 4 \cr 2 & 3 & 4 & 1 \cr } } \right]$$.
Let $$\mathrm{det}(A)$$ and $$\mathrm{det}(B)$$ denote the determinates of the matrices A and B, respectively.
Which one of the options given below is TRUE?
Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.
Let $$\lambda_1,\lambda_2,\lambda_3,\lambda_4$$, and $$\lambda_5$$ be the five eigenvalues of A. Note that these eigenvalues need not be distinct.
The value of $$\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5=$$ ______________
Consider the following two statements with respect to the matrices Am $$\times$$ n , Bn $$\times$$ m , Cn$$\times$$ n and Dn $$\times$$ n .
Statement 1 : tr(AB) = tr(BA)
Statement 2 : tr(CD) = tr(DC)
where tr( ) represents the trace of a matrix. Which one of the following holds?
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?
Where $$A = \left[ {{a_1},.....,\,\,{a_n}} \right]$$ and $$b = \sum\limits_{i = 1}^n {{a_i}.} $$
The set of equations has
Then the rank of $$P+Q$$ is _______.
$$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solution
$$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a solution
$$III.$$ $$\,\,\,$$ If $$m = n,$$ then there exists a system which has a solution
Which one of the following is CORRECT?
3x + 2y = 1
4x + 7z = 1
x + y + z =3
x - 2y + 7z = 0
The number of solutions for this system is ______________________
$$\left| {\matrix{ 1 & x & {{x^2}} \cr 1 & y & {{y^2}} \cr 1 & z & {{z^2}} \cr } } \right|?$$
If the eigen values of $$A$$ are $$4$$ and $$8$$ then
$${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$
$${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$
$${x_1}\, + \,4{x_2}\, + a{x_3}\, = 4$$ has a unique solution. The only possible value (s) for $$\alpha $$ is/are
S1: The sum of two singular n x n matrices may be non-singular
S2: The sum of two n x n non-singular matrices may be singular
Which of the following statements is correct?
x + 2y = 5
4x + 8y = 12
3x + 6y + 3z = 15 This set
two matrices such that $$AB=1.$$
Let $$C = A\left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]$$ and $$CD=1.$$
Express the elements of $$D$$ in terms of the elements of $$B.$$
$$\left[ {\matrix{ 0 & 0 & \alpha \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right],\alpha \ne 0$$ is (are)
calculated by the use of Cayley - Hamilton theoram (or) otherwise is
Marks 2
Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where
$$L=\left[\begin{array}{ccc} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}\right] \text { and } u=\left[\begin{array}{ccc} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}\right]$$
Which of the following statement(s) is/are TRUE?
Let $A$ be a $2 \times 2$ matrix as given.
$$A=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$$
What are the eigenvalues of the matrix $A^{13}$ ?
Let A be an n × n matrix over the set of all real numbers ℝ. Let B be a matrix obtained from A by swapping two rows. Which of the following statements is/are TRUE?
Let A be any n x m matrix, where m > n. Which of the following statements is/are TRUE about the system of linear equations Ax = 0?
Which one of the following is the closed form for the generating function of the sequence (an}n $$\ge$$ 0 defined below?
$${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr {1,} & {otherwise} \cr } } \right.$$
Consider solving the following system of simultaneous equations using LU decomposition.
x1 + x2 $$-$$ 2x3 = 4
x1 + 3x2 $$-$$ x3 = 7
2x1 + x2 $$-$$ 5x3 = 7
where L and U are denoted as
$$L = \left( {\matrix{ {{L_{11}}} & 0 & 0 \cr {{L_{21}}} & {{L_{22}}} & 0 \cr {{L_{31}}} & {{L_{32}}} & {{L_{33}}} \cr } } \right),\,U = \left( {\matrix{ {{U_{11}}} & {{U_{12}}} & {{U_{13}}} \cr 0 & {{U_{22}}} & {{U_{23}}} \cr 0 & 0 & {{U_{33}}} \cr } } \right)$$
Which one of the following is the correct combination of values for L32, U33, and x1 ?
Which of the following is/are the eigenvector(s) for the matrix given below?
$$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} & { - 6} & { - 3} & { - 1} \cr {20} & {15} & 8 & 5 \cr {32} & {21} & 7 & {12} \cr } } \right)$$
For two n-dimensional real vectors P and Q, the operation s(P, Q) is defined as follows:
$$s\left( {P,\;Q} \right) = \mathop \sum \limits_{i = 1}^n \left( {p\left[ i \right].Q\left[ i \right]} \right)$$
Let L be a set of 10-dimensional non-zero vectors such that for every pair of distinct vectors P, Q ∈ L, s(P, Q) = 0. What is the maximum cardinality possible for the set L ?
Consider the following matrix.
$$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{array}} \right)$$
The largest eigenvalue of the above matrix is ______
I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) $$ \le $$ rank(A) + rank(B)
IV. det(A + B) $$ \le $$ det(A) + det(B)
Which of the above statements are TRUE?
Consider the following matrix :
$$ R=\left[\begin{array}{cccc} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{array}\right] $$
The absolute value of the product of Eigen values of $R$ is ___________.
Consider the following statements.
$$\left( {\rm I} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ does not have an inverse
$$\left( {\rm II} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ has a repeated eigenvalue
$$\left( {\rm III} \right)$$ $$\,\,\,\,\,\,\,\,\,$$ $$P$$ cannot be diagonalized
Which one of the following options is correct?
Consider the following statements.
$$(I)$$ One eigenvalue must be in $$\left[ { - 5,5} \right]$$
$$(II)$$ The eigenvalue with the largest magnitude must be strictly greater than $$5$$
Which of the above statements about engenvalues of $$A$$ is/are necessarily correct?
Then which one of the following Options is TRUE?
$$A = \left( {\matrix{ 1 & 4 \cr b & a \cr } } \right)$$
(i) Add the third row to the second row
(ii) Subtract the third column from the first column.
The determinant of the resultant matrix is _________.
the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 & 1 \cr 0 & 1 & 1 & 1 & 0 \cr 0 & 1 & 1 & 1 & 0 \cr 0 & 1 & 1 & 1 & 0 \cr 1 & 0 & 0 & 0 & 1 \cr } } \right]$$ is ____ .
Which of the following options provides the Correct values of the Eigen values of the matrix?
$$S1$$ : Each row of $$M$$ can be represented as a linear combination of the other rows
$$S2$$ : Each column of $$M$$ can be represented as a linear combination of the other columns
$$S3$$ : $$MX$$ $$=$$ $$0$$ has a nontrivial solution
$$S4$$ : $$M$$ has an inverse
$$\left[ {\matrix{ 1 & 0 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matrix{ 0 & 1 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matrix{ 1 & { - 1} \cr 1 & 1 \cr } } \right]\,\,and\,\,\left[ {\matrix{ { - 1} & 0 \cr 1 & { - 1} \cr } } \right]$$
Which of the following is an eigen value of $$\left[ {\matrix{ {\rm A} & {\rm I} \cr {\rm I} & {\rm A} \cr } } \right]$$, where $$I$$ is the $$4$$ $$x$$ $$4$$ identity matrix?
Which one of the following statements is false?
Where $$a, b, c, d, e$$ and $$f$$ are real numbers and $$abc$$ $$ \ne \,\,0$$. Under the matrix multiplication operation, the set $$H$$ is:
$$2x1 - x2 + 3x3 = 1$$
$$3x1 + 2x2 + 5x3 = 2$$
$$ - x1 + 4x2 + x3 = 3$$
This system of equations has
and $${X^2} - X + 1 = 0$$
($${\rm I}$$ is the identity matrix and $$O$$ is the zero matrix), then the inverse of $$X$$ is

What is the value of the determinant of $$A$$?
- x + 5y = - 1
x - y = 2
x + 3y = 3
Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of $$\alpha $$, does this system of equations have infinitely many solutions?
Which of the following is a factor of $$\Delta $$ ?
$$\left[ {\matrix{ a & 0 \cr 0 & b \cr } } \right]\,$$ commute under multiplication