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Let, $A$ and $B$ are square matrices of order $2$ such that $A + a d j (B^{T}) = [\begin{array}{l} 3 & 2 \\ 2 & 3 \end{array}]$ and $A^{T} - a d j (B) = [\begin{array}{l} - 2 & - 1 \\ - 1 & - 2 \end{array}]$ , then $A^{2} + 2 A^{3} + 3 A^{4} + 5 A^{5}$ is equal to (where, $M^{T}$ and $a d j (M)$ represent the transpose matrix and adjoint matrix of matrix $M$ respectively and $I$ represents the identity matrix of order $2$ )

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Important Questions on Determinants

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

If matrix

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

and

A^{- 1} = \frac{1}{k} a d j (A),

then

k

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} c o s θ & - s i n θ \\ s i n θ & c o s θ \end{matrix}]

, then the matrix

A^{- 50}

when

θ = \frac{π}{12},

is equal to:

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 1 + 2 i & i \\ - i & 1 - 2 i \end{matrix}]

, where

i = \sqrt{- 1}

, then

A (a d j A)

= ….

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

If the value of a third order determinant is

16

, then the value of the determinant formed by replacing each of its elements by its cofactor is

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

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

, then

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

is equal to:

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

The inverse of matrix is

[\begin{matrix} 2 & 5 & 0 \\ 0 & 1 & 1 \\ - 1 & 0 & 3 \end{matrix}]

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

Let

A

be any

3 \times 3

invertible matrix. Then which one of the following is not always true?

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is an

3 \times 3

non- singular matrix, such that

A A^{'} = A^{'} A

and

B = A^{- 1} A^{'}

, then

B B^{'}

equals to {where

A^{'}

transpose of matrix

A

}

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

For an invertible matrix

A

A (a d j A) = [\begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix}]

then

| A | =

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix}]

,

then the sum of all the diagonal entries of

A^{- 1}

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A^{- 1} = \frac{1}{11} [\begin{matrix} - 3 & 4 \\ 5 & - 3 \end{matrix}]

, then

A =

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is a

3 \times 3

matrix such that

|5 a d j A| = 5

, then

|A|

is equal to

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

If the matrix

[\begin{matrix} 1 & 2 & - 1 \\ - 3 & 4 & k \\ - 4 & 2 & 6 \end{matrix}]

is singular, then the value of

k

is equal to

HARD

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A = [\begin{matrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & - 1 & 3 \end{matrix}], B = a d j A

and

C = 3 A,

then

\frac{|a d j B|}{|C|}

is equal to

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

Let

A

be a

3 \times 3

matrix such that

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

and

B = adj (a d j A)

. If

| A | = λ

and

|{(B^{- 1})}^{T}| = μ

, then the ordered pair

(|λ|, μ)

is equal to

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

(\begin{array}{l} 2 & 1 \\ 3 & 2 \end{array}) A = (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array})

, then the matrix

A

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

A

is a square matrix of order

3

and

| A | = 5

, then

|A \cdot a d j A|

EASY

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

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

and

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

,

then

x =

MEDIUM

Mathematics>Algebra>Determinants>Adjoint and Inverse of a Matrix

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>Determinants>Adjoint and Inverse of a Matrix

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

Let, A and B are square matrices of order 2 such that A+adjBT=3223 and AT-adjB=-2-1-1-2, then A2+2A3+3A4+5A5 is equal to (where, MT and adj(M) represent the transpose matrix and adjoint matrix of matrix M respectively and I represents the identity matrix of order 2)

Important Questions on Determinants

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