Find minors and cofactors of the elements a 11 , a 21 in the determinant △ = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

By definition of minors and cofactors, we have Minor of a 11 = M 11 = a 22 a 23 a 32 a 33 = a 22 a 33 - a 23 a 32 Cofactor of a 11 = A 11 = - 1 1 + 1 M 11 = a 22 a 33 - a 23 a 32 Minor of a 21 = M 21 = a 12 a 13 a 32 a 33 = a 12 a 33 - a 13 a 32 Cofactor of a 21 = A 21 = - 1 2 + 1 M 21 = - 1 a 12 a 33 - a 13 a 32 = - a 12 a 33 + a 13 a 32 Hence, the required answer is shown above.

Find minors and cofactors of the elements a 11 , a 21 in the determinant △ = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

By definition of minors and cofactors, we have Minor of a 11 = M 11 = a 22 a 23 a 32 a 33 = a 22 a 33 - a 23 a 32 Cofactor of a 11 = A 11 = - 1 1 + 1 M 11 = a 22 a 33 - a 23 a 32 Minor of a 21 = M 21 = a 12 a 13 a 32 a 33 = a 12 a 33 - a 13 a 32 Cofactor of a 21 = A 21 = - 1 2 + 1 M 21 = - 1 a 12 a 33 - a 13 a 32 = - a 12 a 33 + a 13 a 32 Hence, the required answer is shown above.

If Δ 1 = x s i n θ c o s θ - s i n θ - x 1 c o s θ 1 x and Δ 2 = x s i n 2 θ c o s 2 θ - s i n 2 θ - x 1 c o s 2 θ 1 x , x ≠ 0 ; then for all θ ∈ 0 , π 2 :

Δ 1 + Δ 2 = - 2 ( x 3 + x - 1 )

Δ 1 - Δ 2 = x ( c o s 2 θ - c o s 4 θ )

EASY

Earn 100

Find minors and cofactors of the elements $a_{11}, a_{21}$ in the determinant

$△ = |\begin{array}{l} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}|$

Important Questions on Determinants

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

The remainder when the value of the determinant $|\begin{array}{l} 2014^{2014} & 2015^{2015} & 2016^{2016} \\ 2017^{2017} & 2018^{2018} & 2019^{2019} \\ 2020^{2020} & 2021^{2021} & 2022^{2022} \end{array}|$ is divided by $5$ is

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

If the inverse of the matrix

[\begin{matrix} α & 14 & - 1 \\ 2 & 3 & 1 \\ 6 & 2 & 3 \end{matrix}]

does not exist then the value of

α

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

S = \{x \in [0, 2 π] : |\begin{matrix} 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{matrix}| = 0\},

then

\sum_{x \in S} \tan (\frac{π}{3} + x)

is equal to:

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

Which of the following values of

α

satisfy the equation

|\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix}| = - 648 α

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

Let P be a matrix of order

3 \times 3

such that all the entries in P are from the set

\{- 1, 0, 1\}

. Then, the maximum possible value of the determinant of P is _______.

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

The number of distinct real roots of the equation,

|\begin{matrix} \cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\ \sin x & \sin x & \cos x \end{matrix}| = 0

in the interval

[- \frac{π}{4}, \frac{π}{4}]

is :

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

|\begin{matrix} x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6 \end{matrix}| = a x^{4} + b x^{3} + c x^{2} + d x + e

,

then

5 a + 4 b + 3 c + 2 d + e

is equal to

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

The sum of the real roots of the equation

|\begin{matrix} x & - 6 & - 1 \\ 2 & - 3 x & x - 3 \\ - 3 & 2 x & x + 2 \end{matrix}| = 0,

is equal to:

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

|\begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix}| = a x - 12

, then

a

is equal to:

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

The least value of the product

x y z

(such that

x, y and z

are positive real numbers) for which the determinant

|\begin{matrix} x \\ 1 \\ 1 \end{matrix} \begin{matrix} 1 \\ y \\ 1 \end{matrix} \begin{matrix} 1 \\ 1 \\ z \end{matrix}|

is non-negative is

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

Let the numbers

2, b, c

be in an A.P. and

A = [\begin{matrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2} \end{matrix}]

. If

d e t (A) \in [2,16],

then

c

lies in the interval:

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

Let

A = [\begin{matrix} 2 & b & 1 \\ b & b^{2} + 1 & b \\ 1 & b & 2 \end{matrix}],

where

b > 0

. Then the minimum value of

\frac{\det (A)}{b}

is:

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

Let

S

be the sample space of all

3 \times 3

matrices with entries from the set

\{0, 1\} .

Let the events

E_{1}

and

E_{2}

be given by

E_{1} = \{A \in S : d e t A = 0\}

and

E_{2} = {A \in S :

sum of entries of

A

7} .

If a matrix is chosen at random from

S,

then the conditional probability

P (E_{1}| E_{2})

equals____

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

|\begin{matrix} x - 4 & 2 x & 2 x \\ 2 x & x - 4 & 2 x \\ 2 x & 2 x & x - 4 \end{matrix}| = (A + B x) {(x - A)}^{2},

then the ordered pair

(A, B)

is equal to

MEDIUM

Mathematics>Algebra>Determinants>Minor and Cofactors

Δ_{1} = |\begin{matrix} x & s i n θ & c o s θ \\ - s i n θ & - x & 1 \\ c o s θ & 1 & x \end{matrix}|

and

Δ_{2} = |\begin{matrix} x & s i n 2 θ & c o s 2 θ \\ - s i n 2 θ & - x & 1 \\ c o s 2 θ & 1 & x \end{matrix}|,

x \neq 0;

then for all

θ \in (0, \frac{π}{2})

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

The constant term in the expansion of

|\begin{matrix} 3 x + 1 & 2 x - 1 & x + 2 \\ 5 x - 1 & 3 x + 2 & x + 1 \\ 7 x - 2 & 3 x + 1 & 4 x - 1 \end{matrix}|

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

There are two values of

a

which makes determinant,

Δ = |\begin{matrix} 1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2 a \end{matrix}| = 86

, then sum of these number is

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

A = [\begin{matrix} \begin{matrix} 1 & 0 & 2 \\ 5 & 1 & x \\ 1 & 1 & 1 \end{matrix} \end{matrix}]

is a singular matrix, then the value of

x

HARD

Mathematics>Algebra>Determinants>Minor and Cofactors

A = [\begin{matrix} 1 & \sin θ & 1 \\ - \sin θ & 1 & \sin θ \\ - 1 & - \sin θ & 1 \end{matrix}]

, then for all

θ \in (\frac{3 π}{4}, \frac{5 π}{4})

\det (A)

lies in the interval :

EASY

Mathematics>Algebra>Determinants>Minor and Cofactors

Let

d \in R,

and

A = [\begin{matrix} - 2 & 4 + d & (\sin θ) - 2 \\ 1 & (\sin θ) + 2 & d \\ 5 & (2 \sin θ) - d & (- \sin θ) + 2 + 2 d \end{matrix}],

θ \in [0, 2 π] .

If the minimum value of

d e t (A)

8,

then a value of

d

is:

Find minors and cofactors of the elements a11, a21 in the determinant △=a11a12a13a21a22a23a31a32a33

Important Questions on Determinants

Learn and Prepare for any exam you want !

Find minors and cofactors of the elements $a_{11}, a_{21}$ in the determinant

$△ = |\begin{array}{l} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}|$