Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

MEDIUM

Earn 100

Find the rank of the matrix $A = [\begin{matrix} 1 & - 1 & 2 & - 3 \\ 4 & 1 & 0 & 2 \\ 0 & 3 & 0 & 4 \\ 0 & 1 & 0 & 2 \end{matrix}]$

Important Questions on Application of Matrices and Determinants

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

For what values of

μ

the system of homogeneous equations

x + y + 3 z = 0; 4 x + 3 y + μ z = 0; 2 x + y + 2 z = 0

have : only trivial solution.

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

If

A = [\begin{matrix} \sqrt{2020} & \sqrt{2021} & \sqrt{2022} & \sqrt{2023} \\ \sqrt{4040} & \sqrt{4042} & \sqrt{4044} & \sqrt{4046} \\ \sqrt{6060} & \sqrt{6063} & \sqrt{6066} & \sqrt{6069} \\ \sqrt{8080} & \sqrt{8084} & \sqrt{8088} & \sqrt{8092} \end{matrix}]

then the rank of

A

is

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

For what values of

μ

the system of homogeneous equations

x + y + 3 z = 0; 4 x + 3 y + μ z = 0; 2 x + y + 2 z = 0

have : infinitely many solutions.

EASY

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

What is the minimum number of elementary operations that are needed to transform

A = [\begin{matrix} 0 & 1 \\ 1 & 2 \end{matrix}]

to the identity matrix?

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

Let

l, m, n \in R

and

A = [\begin{matrix} 1 & r & r^{2} & l \\ r & r^{2} & 1 & m \\ r^{2} & 1 & r & n \end{matrix}]

. Then the set of all real values of

r

for which the rank of

A

is

3

, is

EASY

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

If the rank of the matrix

(\begin{matrix} λ & - 1 & 0 \\ 0 & λ & - 1 \\ - 1 & 0 & λ \end{matrix})

is

2

, then

λ

is

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

Prove that

ρ (A) + ρ (B) \neq ρ (A + B)

by giving the suitable matrices

A

and

B

of order

3

.

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

The rank of the matrix

[\begin{matrix} - 1 & 2 & 5 \\ 2 & - 4 & a - 4 \\ 1 & - 2 & a + 1 \end{matrix}]

is

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

Suppose

n > 1

and

A

is a non-singular matrix of order

n

such that

|a d j A| = |a d j (a d j A)|

. Then the matrix whose rank is

n

, is

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

The rank of the matrix

[\begin{matrix} 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ 10 & 11 & 12 & 13 & 14 \end{matrix}]

is equal to

EASY

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

Which of the following matrices can NOT be obtained from the matrix

[\begin{matrix} - 1 & 2 \\ 1 & - 1 \end{matrix}]

by a single elementary row operation?

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

For which of the following ordered pairs

(μ, δ),

the system of linear equations

x + 2 y + 3 z = 1

3 x + 4 y + 5 z = μ

4 x + 4 y + 4 z = δ

is inconsistent?

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

The rank of the matrix

[\begin{matrix} 4 & 2 & (1 - x) \\ 5 & k & 1 \\ 6 & 3 & (1 + x) \end{matrix}]

is

1,

then

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

Find the rank of the matrix

[\begin{matrix} 1 & 4 & - 1 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{matrix}] .

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

If the system of linear equations,

$x + y + z = 6$

$x + 2 y + 3 z = 10$

$3 x + 2 y + λ z = μ$

has more than two solutions, then $μ - λ^{2}$ , is equal to.

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

The system of equation

3 x - y + 4 z = 3

x + 2 y - 3 z = - 2

6 x + 5 y + λ z = - 3

Has at least one solution, if

MEDIUM

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

If the matrix

A = [\begin{matrix} \begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 3 \\ 3 & 2 & 1 \end{matrix} \begin{matrix} 0 \\ 2 \\ 3 \end{matrix} \\ \begin{matrix} 6 & 8 & 7 \end{matrix} α \end{matrix}]

is of rank

3

, then

α

equals to

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

The rank of the matrix

[\begin{matrix} 4 & 2 & (1 - x) \\ 5 & k & 1 \\ 6 & 3 & (1 + x) \end{matrix}]

is

2,

then

HARD

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

If the points(

x_{1}, y_{1}), (x_{2}, y_{2})

and

(x_{3}, y_{3})

are collinear,

then the rank of the matrix

[\begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix}]

will always be less than

EASY

Mathematics>Algebra>Application of Matrices and Determinants>Elementary Transformations of a Matrix

The value of

‘ a ’

for which system of equation

a x + y + z = 0, x + a y + z = 0

&

x + y + z = 0

possesses non-null solution is -