MEDIUM

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If a singular matrix $A = {[a_{i j}]}_{2 \times 2}$ always commute with $B = [\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}]$ such that ${(\frac{a_{11}}{a_{12}})}^{2} = k$ , then $k$ is

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Important Questions on Matrices and Determinants

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

Let

f (x) = |\begin{matrix} \sin^{2} x & - 2 + \cos^{2} x & \cos 2 x \\ 2 + \sin^{2} x & \cos^{2} x & \cos 2 x \\ \sin^{2} x & \cos^{2} x & 1 + \cos 2 x \end{matrix}|, x \in [0, π] .

Then the maximum value of

f (x)

is equal to

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

a, b, c

are in

A . P .

then the value of the determinant

|\begin{matrix} x + 2 & x + 3 & x + 2 a \\ x + 3 & x + 4 & x + 2 b \\ x + 4 & x + 5 & x + 2 c \end{matrix}|

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

s = p + q + r,

then value of

|\begin{matrix} s + r & p & q \\ r & s + p & q \\ r & p & s + q \end{matrix}|

is ...........

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

If the system of equations $x + 2 y + 3 z = 3$ , $4 x + 3 y - 4 z = 4$ and $8 x + 4 y - λ z = 9 + μ$ has infinitely many solutions, then the ordered pair $(λ, μ)$ is equal to

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

Δ_{r} = |\begin{matrix} r & 2 r - 1 & 3 r - 2 \\ \frac{n}{2} & n - 1 & a \\ \frac{1}{2} n (n - 1) & {(n - 1)}^{2} & \frac{1}{2} (n - 1) (3 n + 4) \end{matrix}|

, then the value of

\sum_{r = 1}^{n - 1} Δ_{r}

EASY

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

The equation

|\begin{matrix} 1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{matrix}| = 0

has

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

|\begin{matrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{matrix}| =______

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

A

is a

3 \times 3

matrix such that

|5 a d j A| = 5

, then

|A|

is equal to

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

If for some

c < 0,

the quadratic equation,

2 c x^{2} - 2 (2 c - 1) x + 3 c^{2} = 0

has two distinct real roots,

\frac{1}{a}

and

\frac{1}{b},

then the value of the determinant,

|\begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix}|

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

ω

is a complex cube root of unity, then the matrix

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

is a

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

Let

f (x) = |\begin{matrix} 1 + \sin^{2} x & \cos^{2} x & \sin 2 x \\ \sin^{2} x & 1 + \cos^{2} x & \sin 2 x \\ \sin^{2} x & \cos^{2} x & 1 + \sin 2 x \end{matrix}|, x \in [\frac{π}{6}, \frac{π}{3}]

. If

α

and

β

respectively are the maximum and the minimum values of

f

, then

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

a_{1}, a_{2}, \dots . ., a_{n}, \dots .

are in G.P. then

|\begin{matrix} l o g a_{n} & l o g a_{n + 1} & l o g a_{n + 2} \\ l o g a_{n + 3} & l o g a_{n + 4} & l o g a_{n + 5} \\ l o g a_{n + 6} & l o g a_{n + 7} & l o g a_{n + 8} \end{matrix}|

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

Let

D_{k} = |\begin{matrix} \begin{matrix} 1 & 2 k & 2 k - 1 \\ n & n^{2} + n + 2 & n^{2} \\ n & n^{2} + n & n^{2} + n + 2 \end{matrix} \end{matrix}|

. If

\sum_{k = 1}^{n} D_{k} = 96

, then

n

is equal to _________.

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

Let

ω

be the complex number

\cos \frac{2 π}{3} + i \sin \frac{2 π}{3}

. Then the number of distinct complex numbers

z

satisfying

|\begin{matrix} z + 1 & ω & ω^{2} \\ ω & z + ω^{2} & 1 \\ ω^{2} & 1 & z + ω \end{matrix}| = 0

is equal to

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

P = [\begin{matrix} 1 & α & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{matrix}]

is the adjoint of a

3 \times 3

matrix

A

and

|A| = 4

, then

α

is equal to

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

A = [\begin{matrix} 2 & - 3 \\ - 4 & 1 \end{matrix}]

, then

Adj (3 A^{2} + 12 A)

is equal to:

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

The characteristic equation of a matrix

A

λ^{3} - 5 λ^{2} - 3 λ + 2 I = 0

then

|a d j A|

is equal to

MEDIUM

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

a, b, c

are different and

|\begin{array}{l} a & a^{2} & a^{3} - 1 \\ b & b^{2} & b^{3} - 1 \\ c & c^{2} & c^{3} - 1 \end{array}| = 0

, then

a b c

is equal to

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

α, β \neq 0

f (n) = α^{n} + β^{n}

and

|\begin{matrix} 3 & 1 + f (1) & 1 + f (2) \\ 1 + f (1) & 1 + f (2) & 1 + f (3) \\ 1 + f (2) & 1 + f (3) & 1 + f (4) \end{matrix}| = K {(1 - α)}^{2} {(1 - β)}^{2} {(α - β)}^{2}

, then

K

is equal to

HARD

Mathematics>Algebra>Matrices and Determinants>Properties of Determinants

Let

a_{1}, a_{2}, a_{3} \dots, a_{10}

be in

G . P .

with

a_{i} > 0

for

i = 1, 2, \dots, 10

and

S

be the set of pairs

(r, k), r, k \in N

(the set of natural numbers) for which

|\begin{matrix} \log_{e} a_{1}^{r} a_{2}^{k} & \log_{e} a_{2}^{r} a_{3}^{k} & \log_{e} a_{3}^{r} a_{4}^{k} \\ \log_{e} a_{4}^{r} a_{5}^{k} & \log_{e} a_{5}^{r} a_{6}^{k} & \log_{e} a_{6}^{r} a_{7}^{k} \\ \log_{e} a_{7}^{r} a_{8}^{k} & \log_{e} a_{8}^{r} a_{9}^{k} & \log_{e} a_{9}^{r} a_{10}^{k} \end{matrix}| = 0

Then the number of elements in

S,

is:

If a singular matrix A=aij2×2 always commute with B=1121 such that a11a122=k, then k is

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

If a singular matrix $A = {[a_{i j}]}_{2 \times 2}$ always commute with $B = [\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}]$ such that ${(\frac{a_{11}}{a_{12}})}^{2} = k$ , then $k$ is