GATE CS Linear Algebra: Matrices, Determinants, System of Linear Equations questions

Practise Linear Algebra: Matrices, Determinants, System of Linear Equations questions for GATE CS. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

Start with matrix dimensions and the conditions for each operation. Check rank, eigenvalue and invertibility assumptions before applying a theorem.

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SubjectTopicYearType

n>1

n \times n

M

with its elements from

\mathbb{R}

. Let the vector

(0,1,0,0, \ldots, 0) \in \mathbb{R}^{n}

be in the null space of

M

. Which of the following options is/are always correct?

Consider the system of linear equations given below.

\begin{array}{c} a x+y=b \\ 16 x+a y=24 \end{array}

Suppose the values of

a

b

are chosen such that the system of linear equations produce multiple solutions. Then the product of

a

b

\_\_\_\_

. (answer in integer)

The determinant of a

4 \times 4

A

3

. The value of the determinant of

2 A

\_\_\_\_

. (answer in integer)

Consider the given system of linear equations for variables

x

y

k

is a real-valued constant. Which of the following option(s) is/are CORRECT?

\begin{array}{l} x+k y=1 \\ k x+y=-1 \end{array}

A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)

, then which ONE of the following is

A^{8}

L

M

N

be non-singular matrices of order

3

satisfying the equations

L^2 = L^{-1}

M=L^8

N=L^2

. Which ONE of the following is the value of the deteminant of

(M-N)

Consider a system of linear equations

P X=Q

P \in \mathbb{R}^{3 \times 3}

\mathrm{Q} \in \mathbb{R}^{3 \times 1}

P

has an LU decomposition,

P=L U

A

n \times m

m>n

. Which of the following statements is/are TRUE about the system of linear equations

Ax=0

A

n \times n

matrix over the set of all real numbers

\mathbb{R}

B

be a matrix obtained from

A

by swapping two rows. Which of the following statements is/are TRUE?

A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{array}\right]

B=\left[\begin{array}{llll} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{array}\right]

Consider the following two statements with respect to the matrices

\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}

\textit{D}_{n \times n}.

1: tr \text{(AB)} = tr \text{(BA)}

2: tr \text{(CD)} = tr \text{(DC)}

tr()

represents the trace of a matrix. Which one of the following holds?

Consider solving the following system of simultaneous equations using

\text{LU}

x_{1} + x_{2} – 2x_{3} = 4

x_{1} + 3x_{2} – x_{3} = 7

2x_{1} + x_{2} – 5x_{3} = 7

\textit{L}

\textit{U}

are denoted as …

P

4 \times 5

matrix such that every solution of the equation

\text{Px=0}

is a scalar multiple of

\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T

P

A

B

n \times n

matrices over real numbers. Let rank(

M

\text{det}(M)

denote the rank and determinant of a matrix

M

, respectively. Consider the following statements.

\text{rank}(AB) = \text{rank }(A) \text{rank }(B)

\text{det}(AB) = \text{det}(A) \text{det}(B)

X

be a square matrix. Consider the following two statements on

X

X

is invertible Determinant of

X

is non-zero Which one of the following is TRUE?

p

q

r

be propositions and the expression

\left ( p\rightarrow q \right )\rightarrow r

be a contradiction. Then, the expression

\left ( r\rightarrow p \right )\rightarrow q

u

v

be two vectors in

\mathbf{R}^{2}

whose Euclidean norms satisfy

\left \| u \right \| = 2\left \| v \right \|

. What is the value of

\alpha

w = u + \alpha v

bisects the angle between

u

v

c_{1}.....c_{n}

be scalars, not all zero, such that

\sum_{i=1}^{n}c_{i}a_{i}

a_{i}

are column vectors in

R^{n}

. Consider the set of linear equations

Ax = b

A=\left [ a_{1}.....a_{n} \right ]

b=\sum_{i=1}^{n}a_{i}

. The set of equations has

P = \begin{bmatrix}1 & 1 & -1 \\2 & -3 & 4 \\3 & -2 & 3\end{bmatrix}

Q = \begin{bmatrix}-1 & -2 &-1 \\6 & 12 & 6 \\5 & 10 & 5\end{bmatrix}

be two matrices. Then the rank of

P+Q

is ___________ .

Consider the systems, each consisting of

m

linear equations in

n

m < n

, then all such systems have a solution. If

m > n

, then none of these systems has a solution. If

m = n

, then there exists a system which has a solution. Which one of the following is CORRECT?