Matrices and Determinants jee mains questions | JEE Main Maths Matrices and Determinants Previous Year Question

Q 11 :

Let A be a $3 \times 3$ matrix such that $A + A^{T} = O$ . If $A [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}], A^{2} [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} - 3 \\ 19 \\ - 24 \end{matrix}],$ and
$de t (a dj (2 adj (A + I))) = {(2)}^{α} \cdot {(3)}^{β} \cdot {(11)}^{γ},$ $α, β, γ$ are non-negative integers, then $α + β + γ$ is equal to _________ . [2026]

(18)

$A + A^{T} = O$

$A [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}] and A^{2} [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} - 3 \\ 19 \\ - 24 \end{matrix}]$

$A + A^{T} = O$

$A is skew symmetric matrix$

$A = [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}]$

Now ATQ

$[\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}]$

$a = - 3 and c = b + 2$

$A^{2} [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = A [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}] = [\begin{matrix} 0 & - 3 & b \\ 3 & 0 & c \\ - b & - c & 0 \end{matrix}] [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}]$

$= [\begin{matrix} - 9 + 2 b \\ 9 + 2 c \\ - 3 b - 3 c \end{matrix}] = [\begin{matrix} - 3 \\ 19 \\ - 24 \end{matrix}]$

By comparing we get

$b = 3, c = 5$

Now $A = [\begin{matrix} 0 & - 3 & 3 \\ 3 & 0 & 5 \\ - 3 & - 5 & 0 \end{matrix}]$

Let $B = A + I = [\begin{matrix} 1 & - 3 & 3 \\ 3 & 1 & 5 \\ - 3 & - 5 & 1 \end{matrix}]$

$\det (B) = 44 = 2^{2} \times 11$

we find $D = \det (adj (2 adj (B)))$

$D = 2^{14} \cdot 11^{4}$

Compare with $2^{α} 3^{β} 11^{γ}$

$α = 14, β = 0, γ = 4$

$α + β + γ = 18$

Q 12 :

If $Δ =$ $| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} |$ $,$ then the cofactor $A_{21}$ is:

$- (h c + f g)$
$f g - h c$
$f g + h c$
$h c - f g$

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(2)

Ans. $f g - h c$

Explanation:

$A_{21} = {(- 1)}^{2 + 1} M_{21}$

$= - M_{21} = -$ $| \begin{matrix} h & g \\ f & c \end{matrix} |$

$= - (h c - f g)$

$= f g - h c$

Q 13 :

If $Δ =$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$ and $A_{i j}$ is the cofactor of $a_{i j}$ , then the value of $Δ$ is given by:

$a_{11} A_{31} + a_{12} A_{32} + a_{13} A_{33}$
$a_{11} A_{11} + a_{12} A_{21} + a_{13} A_{31}$
$a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13}$
$a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$

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(4)

Ans. $a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$

Explanation:

$∆ =$ sum of the products of the elements of any row (or any column) with their corresponding cofactors.

$∴$ $Δ = a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$

Q 14 :

If $A = [\begin{matrix} 2 & 3 \\ - 4 & - 6 \end{matrix}],$ then which of the following is true?

$A (adj A) \neq$ $| A |$ $I$
$A (adj A) \neq (adj A) A$
$A (adj A) = (adj A) A =$ $| A |$ $I = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$
None of the above

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(3)

Ans. $A (adj A) = (adj A) A =$ $| A |$ $I = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

Explanation:

We know that if $A$ is any square matrix of order $n$ , then $A (adj A) = (adj A) A =$ $| A |$ $\cdot I .$

Q 15 :

If A and B are invertible matrices, then which of the following is not correct?

$adj A =$ $| A |$ $\cdot A^{- 1}$
$\det {(A)}^{- 1} = {[\det (A)]}^{- 1}$
${(A B)}^{- 1} = B^{- 1} A^{- 1}$
${(A + B)}^{- 1} = B^{- 1} + A^{- 1}$

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(4)

Ans. ${(A + B)}^{- 1} = B^{- 1} + A^{- 1}$

Explanation:

Since A and B are invertible matrices, we can say that

${(A B)}^{- 1} = B^{- 1} A^{- 1}$ $\dots (i)$

We know that,

$A^{- 1} = \frac{1}{| A |} (adj A)$

$\Rightarrow$ $adj A =$ $| A |$ $\cdot A^{- 1}$

Also, $\det {(A)}^{- 1} = {[\det (A)]}^{- 1}$

$\Rightarrow$ $\det {(A)}^{- 1} = \frac{1}{\det (A)}$

$\Rightarrow$ $\det (A) \cdot \det {(A)}^{- 1} = 1,$

which is true.

Q 16 :

If $A = [\begin{matrix} 2 & λ & - 3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{matrix}],$ then $A^{- 1}$ exists if:

$λ = 2$
$λ \neq 2$
$λ = - 2$
None of these

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(4)

Ans. None of these

Explanation:

We have, $A = [\begin{matrix} 2 & λ & - 3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{matrix}]$

$A^{- 1}$ exists if $| A |$ $\neq 0$

Now, $| A |$ $= 2 (6 - 5) - λ (- 5) - 3 (- 2) = 8 + 5 λ$

But, $| A |$ $\neq 0$

$5 λ + 8 \neq 0$ $\Rightarrow 5 λ \neq - 8$

$\Rightarrow$ $λ \neq - \frac{8}{5}$

So, $A^{- 1}$ exists if and only if $λ \neq \frac{- 8}{5}$

Q 17 :

If $A = [\begin{matrix} 2 & 1 & 0 \\ 3 & 1 & 2 \\ 0 & 4 & - 1 \end{matrix}],$ then $| adj (A) |$ is equal to:

11
12
225
- 225

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(3)

Ans. 225

Explanation:

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

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

Now, $A = 2 (- 1 - 8) - 1 (- 3 - 0) + 0$

$= - 18 + 3$

$= - 15$

Since for a $3 \times 3$ matrix,

$| adj (A) |$ $= | A |^{3 - 1} = | A |^{2}$

we get $| adj (A) |$ $= {(- 15)}^{2} = 15^{2} = 225$

Q 18 :

If $A$ and $B$ are invertible matrices of order 3, $| A |$ $= 2$ and $| {(A B)}^{- 1} |$ $= - \frac{1}{6},$ then the value of $| B |$ is:

$3$
$- 3$
$2$
$\frac{1}{6}$

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(2)

Ans. $- 3$

Explanation:

Given, $| A |$ $= 2,$

and $| {(A B)}^{- 1} | = \frac{- 1}{6}$

We know that

$| B B^{- 1} |$ $=$ $I$ and $| A A^{- 1} |$ $= I$

$| B^{- 1} |$ $= \frac{1}{B}$ and $| A^{- 1} | = \frac{1}{| A |} = \frac{1}{2}$

${(A B)}^{- 1} = B^{- 1} A^{- 1}$

$\Rightarrow$ $| B^{- 1} |$ $| A^{- 1} |$ $= \frac{- 1}{6}$

$\Rightarrow$ $\frac{1}{B} \times \frac{1}{2} = \frac{- 1}{6}$

$\Rightarrow$ $B = - 3$

Q 19 :

Let $A = [\begin{matrix} 1 & - 2 & 3 \\ 1 & 2 & 1 \\ λ & 2 & - 3 \end{matrix}] .$ If $A^{- 1}$ does not exist, then $λ =$

$- 2$
$2$
$1$
$- 1$

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(4)

Ans. $- 1$

Explanation:

Given,

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

Since $A^{- 1}$ does not exist $\Rightarrow$ $| A |$ $= 0$

Now, $1 (- 6 - 2) + 2 (- 3 - λ) + 3 (2 - 2 λ) = 0$

$- 8 - 6 - 2 λ + 6 - 6 λ = 0$

$- 8 λ - 8 = 0$

$λ = - 1$

Q 20 :

The sum of the products of elements of any row with the cofactors of corresponding elements is:

the value of the determinant
0
sum of cofactors
adjoint of matrix

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(1)

Ans. the value of the determinant

Explanation:

The sum of the products of the elements of any row with the cofactors of the corresponding elements is equal to the value of the determinant.

Let $Δ =$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$

Q 21 :

If A is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))))) = 2^{m} 3^{n} .$ Then $m + 2 n$ is equal to:

2
3
6
4

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3

4

(4)

$| A |$ $= 3$

$| adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1})))) |$

$=$ $| - 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))) |^{2}$

$= 4^{6}$ $| adj (- 3 adj (3 adj ({(2 A)}^{- 1}))) |^{2}$

$= 2^{12} \cdot 3^{12}$ $| 3 adj {(2 A)}^{- 1} |^{8}$

$= 2^{12} \cdot 3^{12} \cdot 3^{24}$ $| adj {(2 A)}^{- 1} |^{8}$

$= 2^{12} \cdot 3^{36}$ $| {(2 A)}^{- 1} |^{16}$

$= 2^{12} \cdot 3^{36} \frac{1}{| 2 A |^{16}}$

$= 2^{12} \cdot 3^{36} \frac{1}{2^{48} | A |^{16}}$

$= 2^{12} \cdot 3^{36} \frac{1}{2^{48} 3^{16}}$

$= \frac{3^{20}}{2^{36}} = 2^{- 36} \cdot 3^{20}$

$m = - 36,$ $n = 20$

$m + 2 n = 4$

Q 22 :

If A is matrix of order 3 such that $| A |$ $= 5$ and $B = adj A$ , then the value of $| | A^{- 1} | {(A B)}^{T} |$ is equal to (where $| A |$ denotes determinant of matrix $A . A^{T}$ denotes transpose of matrix $A,$ $A^{- 1}$ denotes inverse of matrix $A$ , $adj A$ denotes adjoint of matrix A)

$5$
$1$
$25$
$\frac{1}{25}$

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(2)

$| A^{- 1} |$ $= \frac{1}{| A |} = \frac{1}{5}$

$| {(A B)}^{T} |$ $=$ $| A B |$ $=$ $| A (adj A) |$ $=$ $| A |$ $\cdot$ $| adj A |$ $= 5 \times 5^{2} = 5^{3}$

$∴$ $| | A^{- 1} | {(A B)}^{T} |$ $=$ $| \frac{1}{5} {(A B)}^{T} |$ $= \frac{1}{5^{3}}$ $| A B |$ $= 1$

Q 23 :

If $A$ is a square matrix of order $n$ then, $\underset{(n - 1) times}{\underset{⏟}{(adj adj adj \dots adj A)} \cdot} \underset{n times}{\underset{⏟}{(adj adj adj \dots adj A)}}$ is equal to:

$| A^{n - 1} |^{{(n - 1)}^{2}}$
$| A^{- 1} |^{{(n - 1)}^{(n - 1)}}$
$\frac{1}{n} | A |^{{(n - 1)}^{(n - 1)}} \cdot I_{n \times n}$
$| A |^{{(n - 1)}^{(n - 1)}} \cdot I_{n \times n}$

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(4)

$A \cdot Adj A =$ $| A |$ $I_{n}$

$(Adj A) (AdjAdj A) =$ $| Adj A |$ $I_{n} =$ $| A |^{n - 1}$ $I_{n}$

$(AdjAdj A) (AdjAdjAdj A) =$ $| A |^{{(n - 1)}^{2}}$ $I_{n}$

$\Rightarrow$ $(AdjAdj \dots A) (AdjAdjAdj \dots A) =$ $| A |^{{(n - 1)}^{(n - 1)}}$ $I_{n}$

Q 24 :

If $2 \sin α \sin β + 3 \cos β + 5 \cos α \sin β = \sqrt{38} \forall α, β \in R,$ then $\det (adj (adj A))$ is equal to (where $A = [\begin{matrix} \tan β & - \frac{1}{3} \\ 1 & \sin α \end{matrix}]$ ) is equal to ______

(1)

$Let \vec{p} = 2 \hat{i} + 3 \hat{j} + 5 \hat{k}; \vec{q} = \sin α \sin β \hat{i} + \cos β \hat{j} + \cos α \sin β \hat{k}$

$| \vec{q} |$ $= \sqrt{\sin^{2} α \sin^{2} β + \cos^{2} β + \cos^{2} α \sin^{2} β} = 1$

$\Rightarrow \sin α \sin β = 2 λ; \cos β = 3 λ; \cos α \sin β = 5 λ$

$1 = 38 λ^{2}; λ = \frac{1}{\sqrt{38}}$

$\det A = [\frac{\sin α \sin β}{\cos β} + \frac{1}{3}] = 1$

Q 25 :

Let $P =$ $[\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}]$ , where $α \in R$ . Suppose $Q = [q_{i j}]$ is a matrix satisfying $P Q = k I_{3}$ for some non-zero $k \in R$ . If $q_{23} = - \frac{k}{8}$ and $| Q |$ $= \frac{k^{2}}{2}$ , then $α^{2} + k^{2}$ is equal to

13
15
17
21

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(3)

$Given that P Q = k I$

$| P |$ $.$ $| Q |$ $= k^{3}$

$\Rightarrow$ $| P |$ $= 2 k \neq 0 \Rightarrow P is an invertible matrix$

$∵$ $P Q = k I$ $∴$ $Q = k P^{- 1} I$ $[∵ P^{- 1} P = I]$

$∴$ $Q = \frac{adj P}{2}$ $∵$ $q_{23} = - \frac{k}{8}$

$∴$ $- \frac{(3 α + 4)}{2} = \frac{k}{8} \Rightarrow 12 α + 16 = k$ $. . . (i)$

$∴$ $| P |$ $= 2 k \Rightarrow k = 10 + 6 α$

$From (i) and (ii) we get α = - 1, k = 4$

$∴$ $α^{2} + k^{2} = 17$

Q 26 :

Consider a matrix $A =$ $[\begin{matrix} α & β & λ \\ α^{2} & β^{2} & λ^{2} \\ β + λ & λ + α & α + β \end{matrix}]$ where $α, β, λ$ are three distinct natural numbers.

If $\frac{\det (adj (adj (adj (adj A))))}{{(α - β)}^{16} {(β - γ)}^{16} {(γ - α)}^{16}} = 2^{32} \times 3^{16}$ , then the number of such 3-triples $(α, β, γ)$ is _______ .

(42)

$Given matrix is A =$ $[\begin{matrix} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ β + γ & γ + α & α + β \end{matrix}]$

$R_{3} \to R_{3} + R_{1}$

$\Rightarrow$ $| A |$ $=$ $| α + β + γ |$ $| \begin{matrix} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ 1 & 1 & 1 \end{matrix} |$

$\Rightarrow$ $| A |$ $= (α + β + γ) (α - β) (β - γ) (γ - α)$

$∵$ $| adj A |$ $=$ $| A |^{n - 1}$

$| adj (adj A) |$ $=$ $| A |^{{(n - 1)}^{2}}$

$| adj (adj (adj A)) |$ $=$ $| A |^{{(n - 1)}^{3}}$ $=$ $| A |^{8}$

$| adj (adj (adj (adj A))) |$ $=$ $| A |^{{(n - 1)}^{4}}$ $=$ $| A |^{16}$

$∴$ $(α + β + γ) = 2^{32} \cdot 3^{16}$

Q 27 :

Let $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$ . Then the sum of all elements of the matrix $adj (adj {(2 adj A)}^{- 1}))$ is equal to : [2026]

3
4
–4
–3

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2

3

4

(4)

We have, $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$

|A| = 1(–3) – 1(–10 – 1) + 2(–6) = –3 + 11 – 12 = –4 $\neq$ 0

Let $B = {2 adj A)}^{- 1}$

Then $adj {(adj B) = | B |}^{n - 2} \cdot B$

$= | 2 {(adj A)}^{- 1} | \cdot 2 {(adj A)}^{- 1}$ [ $∵$ n = 1]

$= \frac{2^{3}}{| adj A |} \cdot \frac{A}{| A |} = \frac{2^{4}}{| A |^{2}} \times \frac{A}{| A |} = \frac{2^{4} A}{16 | A |} = \frac{A}{| A |}$

Sum of elements of A = 12

Sum of elements of the matrix adj (adj) B) = $\frac{12}{- 4} = - 3$ .

Q 28 :

Let A be a $3 \times 3$ matrix such that $A^{T} = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 5 \\ 2 \\ 2 \end{matrix}]$ , $A^{T} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 1 \\ 1 \end{matrix}]$ , $A = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 4 \\ 4 \end{matrix}]$ and $A = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 3 \\ 1 \end{matrix}]$ . If det (A) = 1, then $\det (adj (A^{2} + A))$ is equal to : [2026]

16
25
49
64

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2

3

4

(4)

Let $A = [\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}]$

$\Rightarrow A = [\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}] [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 4 \\ 4 \end{matrix}]$

$\begin{matrix} \Rightarrow a_{1} + a_{3} = 3 \\ \Rightarrow a_{4} + a_{6} = 4 \\ \Rightarrow a_{7} + a_{9} = 4 \end{matrix}\}$ (i)

Also, $[\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}] [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ 3 \\ 1 \end{matrix}]$

$\begin{matrix} \Rightarrow a_{3} = 1 \Rightarrow a_{1} = 2 \\ \Rightarrow a_{6} = 3 \Rightarrow a_{4} = 1 \\ \Rightarrow a_{9} = 1 \Rightarrow a_{7} = 3 \end{matrix}\}$ [From (i)]

Now, $[\begin{matrix} a_{1} & a_{4} & a_{7} \\ a_{2} & a_{5} & a_{8} \\ a_{3} & a_{6} & a_{9} \end{matrix}] [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 1 \\ 1 \end{matrix}]$

$\begin{matrix} \Rightarrow a_{7} = 3 \\ \Rightarrow a_{8} = 1 \\ \Rightarrow a_{9} = 1 \end{matrix}\}$ (ii)

Also, $[\begin{matrix} a_{1} & a_{4} & a_{7} \\ a_{2} & a_{5} & a_{8} \\ a_{3} & a_{6} & a_{9} \end{matrix}] [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} 5 \\ 2 \\ 2 \end{matrix}]$

$\begin{matrix} \Rightarrow a_{1} + a_{7} = 5 \Rightarrow a_{1} = 2 \\ \Rightarrow a_{2} + a_{8} = 2 \Rightarrow a_{2} = 1 \\ \Rightarrow a_{3} + a_{9} = 2 \Rightarrow a_{3} = 1 \end{matrix}\}$ [From (ii)]

$| A | = | \begin{matrix} 2 & 1 & 1 \\ 1 & a_{5} & 3 \\ 3 & 1 & 1 \end{matrix} | \Rightarrow - a_{5} + 3 = 1 \Rightarrow a_{5} = 2$

$∴ A = | \begin{matrix} 2 & 1 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} |$

Now, $| adj A \cdot (A + I) | = | adj (A + I) | | adj A |$

$| A + I | = | \begin{matrix} 3 & 1 & 1 \\ 1 & 3 & 3 \\ 3 & 1 & 2 \end{matrix} | = 8$

$∴ | adj (A^{2} + A) | = 64 \times 1 = 64$

Q 29 :

Let $A = [\begin{matrix} α & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{matrix}]$ and $B = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 5 α & 0 \\ 0 & 4 α & - 2 α \end{matrix}] + adj (A)$ . If det (B) = 66, then det (adj (A)) equals : [2026]

289
361
441
529

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(3)

$A = [\begin{matrix} α & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{matrix}]$

$B = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 5 α & 0 \\ 0 & 4 α & - 2 α \end{matrix}] + a d j (A)$

$adj (A) = {[\begin{matrix} 15 & - 10 & 8 \\ 3 & 5 α & - 4 α \\ - 6 & 4 & 3 α - 2 \end{matrix}]}^{T} = [\begin{matrix} 15 & 3 & - 6 \\ - 10 & 5 α & 4 \\ 8 & 4 α & 3 α - 2 \end{matrix}]$

$∴ B = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 5 α & 0 \\ 0 & 4 α & - 2 α \end{matrix}] + [\begin{matrix} 15 & 3 & - 6 \\ - 10 & 5 α & 4 \\ 8 & 4 α & 3 α - 2 \end{matrix}]$

$\Rightarrow B = [\begin{matrix} 16 & 3 & - 6 \\ - 10 & 0 & 4 \\ 8 & 0 & α - 2 \end{matrix}]$

Now, det (B) = 66

$\Rightarrow - 3 (- 10 α + 20 - 32) = 66 \Rightarrow α = 1$

$\det (A) = | \begin{matrix} 1 & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{matrix} | = 15 - 10 + 16 = 21$

$∴ \det (adj A) = | A |^{2} = {(21)}^{2} = 441$

Q 30 :

Consider the matrices $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}]$ and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}]$ . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

(34)

We have, $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}] \Rightarrow | A | = - 4 + 8 = 4$

and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}] \Rightarrow | B | = 9 - 9 = 0$

$∴ A^{- 1} = \frac{1}{| A |} adj (A) = \frac{1}{4} [\begin{matrix} - 2 & 2 \\ - 4 & 2 \end{matrix}]$

Now, PA = B

$\Rightarrow P = B A^{- 1}$

$= [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}] \times \frac{1}{4} [\begin{matrix} - 2 & 2 \\ - 4 & 2 \end{matrix}] = \frac{1}{4} [\begin{matrix} - 42 & 24 \\ - 14 & 8 \end{matrix}]$

Now, $A Q = B \Rightarrow Q = A^{- 1} B$

$= \frac{1}{4} [\begin{matrix} - 2 & 2 \\ - 4 & 2 \end{matrix}] [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}] = \frac{1}{4} [\begin{matrix} - 4 & 12 \\ - 10 & - 30 \end{matrix}]$

Now, sum of diagonal elements of 2(P + Q)

$= (- \frac{46}{4} - \frac{22}{4}) = 2 (- \frac{68}{4}) = - 34$

$∴$ Absolute value = 34.

Q 31 :

Let $A = [\begin{matrix} - 1 & 1 & - 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{matrix}]$ satisfy $A^{2} + α (adj (adj (A))) + β (adj (A) (adj (adj (A)))) = [\begin{matrix} 2 & - 2 & 2 \\ - 2 & 0 & - 1 \\ 0 & 0 & - 1 \end{matrix}]$ for some $α, β \in R$ . Then ${(α - β)}^{2}$ is equal to _______. [2026]

(4)

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

$\Rightarrow | A | = - 1$

$∵ adj (adj A) = | A |^{n - 2} A$

and $A a d j A = | A | I$

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

We have, $A^{2} + α (adj (adj (A)) + β (adj (A) (adj (adj (A))) = [\begin{matrix} 2 & - 2 & 2 \\ - 2 & 0 & - 1 \\ 0 & 0 & - 1 \end{matrix}]$

$\Rightarrow A^{2} + α (- A) + β I = [\begin{matrix} 2 & - 2 & 2 \\ - 2 & 0 & - 1 \\ 0 & 0 & - 1 \end{matrix}]$

$\Rightarrow [\begin{matrix} 2 & - 1 & 1 \\ - 1 & 0 & - 1 \\ 0 & 0 & - 1 \end{matrix}] - α [\begin{matrix} - 1 & 1 & - 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{matrix}] + β = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & - 2 & 2 \\ - 2 & 0 & - 1 \\ 0 & 0 & - 1 \end{matrix}]$

$\Rightarrow - [\begin{matrix} 2 + α + β & - 1 - α & 1 + α \\ - 1 - α & 1 + β & - α \\ 0 & 0 & 1 - α + β \end{matrix}] = [\begin{matrix} 2 & - 2 & 2 \\ - 2 & 0 & - 1 \\ 0 & 0 & - 1 \end{matrix}]$

By comparison on both sides, we get

$α = 1 and 1 + β = 0 \Rightarrow β = - 1$

$∴ {(α - β)}^{2} = {(1 + 1)}^{2} = 2^{2} = 4$

Q 32 :

If $A = [\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix}]$ and $I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ then $20 A^{- 1}$ = ____

$2 A - 12 I$
$A - 6 I$
$A + 6 I$
$2 A + 12 I$

1

2

3

4

(1)

$A^{- 1} = \frac{1}{8 - 18} [\begin{matrix} 4 & - 2 \\ - 9 & 2 \end{matrix}],$ $10 A^{- 1} = [\begin{matrix} - 4 & 2 \\ 9 & - 2 \end{matrix}]$

Q 33 :

A is $3 \times 3$ matrix and $| A |$ $= 4$ then $| a d j (a d j A) |$ $= 4^{k}$ . Then $k$ is _____

(4)

$| adj (adj A) |$ $=$ $| A |^{{(n - 1)}^{2}}$

Topic Question Set

Q 12 :

If $Δ =$ $| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} |$ $,$ then the cofactor $A_{21}$ is:

Q 13 :

If $Δ =$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$ and $A_{i j}$ is the cofactor of $a_{i j}$ , then the value of $Δ$ is given by:

Q 14 :

If $A = [\begin{matrix} 2 & 3 \\ - 4 & - 6 \end{matrix}],$ then which of the following is true?

Q 15 :

If A and B are invertible matrices, then which of the following is not correct?

Q 16 :

If $A = [\begin{matrix} 2 & λ & - 3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{matrix}],$ then $A^{- 1}$ exists if:

Q 17 :

If $A = [\begin{matrix} 2 & 1 & 0 \\ 3 & 1 & 2 \\ 0 & 4 & - 1 \end{matrix}],$ then $| adj (A) |$ is equal to:

Q 18 :

If $A$ and $B$ are invertible matrices of order 3, $| A |$ $= 2$ and $| {(A B)}^{- 1} |$ $= - \frac{1}{6},$ then the value of $| B |$ is:

Q 19 :

Let $A = [\begin{matrix} 1 & - 2 & 3 \\ 1 & 2 & 1 \\ λ & 2 & - 3 \end{matrix}] .$ If $A^{- 1}$ does not exist, then $λ =$

Q 20 :

The sum of the products of elements of any row with the cofactors of corresponding elements is:

Q 21 :

If A is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))))) = 2^{m} 3^{n} .$ Then $m + 2 n$ is equal to:

Q 23 :

If $A$ is a square matrix of order $n$ then, $\underset{(n - 1) times}{\underset{⏟}{(adj adj adj \dots adj A)} \cdot} \underset{n times}{\underset{⏟}{(adj adj adj \dots adj A)}}$ is equal to:

Q 24 :

If $2 \sin α \sin β + 3 \cos β + 5 \cos α \sin β = \sqrt{38} \forall α, β \in R,$ then $\det (adj (adj A))$ is equal to (where $A = [\begin{matrix} \tan β & - \frac{1}{3} \\ 1 & \sin α \end{matrix}]$ ) is equal to ______

Q 27 :

Let $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$ . Then the sum of all elements of the matrix $adj (adj {(2 adj A)}^{- 1}))$ is equal to : [2026]

Q 29 :

Let $A = [\begin{matrix} α & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{matrix}]$ and $B = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 5 α & 0 \\ 0 & 4 α & - 2 α \end{matrix}] + adj (A)$ . If det (B) = 66, then det (adj (A)) equals : [2026]

Q 30 :

Consider the matrices $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}]$ and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}]$ . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

Q 32 :

If $A = [\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix}]$ and $I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ then $20 A^{- 1}$ = ____

Q 33 :

A is $3 \times 3$ matrix and $| A |$ $= 4$ then $| a d j (a d j A) |$ $= 4^{k}$ . Then $k$ is _____

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Topic Question Set

Q 11 : Let A be a 3×3 matrix such that A+AT=O. If A[1-10]=[332], A2[1-10]=[-319-24], and det(adj(2adj(A+I)))=(2)α·(3)β·(11)γ, α,β,γ are non-negative integers, then α+β+γ is equal to _________ . [2026]

Q 12 : If Δ=|ahghbfgfc|, then the cofactor A21 is:

Q 13 : If Δ=|a11a12a13a21a22a23a31a32a33| and Aij is the cofactor of aij, then the value of Δ is given by:

Q 14 : If A=[23-4-6], then which of the following is true?

Q 15 : If A and B are invertible matrices, then which of the following is not correct?

Q 16 : If A=[2λ-3025113], then A-1 exists if:

Q 17 : If A=[21031204-1], then |adj(A)| is equal to:

Q 18 : If A and B are invertible matrices of order 3, |A|=2 and |(AB)-1|=-16, then the value of |B| is:

Q 19 : Let A=[1-23121λ2-3]. If A-1 does not exist, then λ=

Q 20 : The sum of the products of elements of any row with the cofactors of corresponding elements is:

Q 21 : If A is a square matrix of order 3 such that det(A)=3 and det(adj(-4 adj(-3 adj(3 adj((2A)-1)))))=2m3n. Then m+2n is equal to:

Q 22 : If A is matrix of order 3 such that |A|=5 and B=adjA, then the value of ||A-1|(AB)T| is equal to (where |A| denotes determinant of matrix A.AT denotes transpose of matrix A, A-1 denotes inverse of matrix A, adjA denotes adjoint of matrix A)

Q 23 : If A is a square matrix of order n then, (adjadjadj…adjA)⏟·(n-1) times(adjadjadj…adjA)⏟n times is equal to:

Q 24 : If 2sinα sinβ+3cosβ+5cosα sinβ=38 ∀α,β∈R, then det(adj(adjA)) is equal to (where A=[tanβ-131sinα]) is equal to ______

Q 25 : Let P=[3-1-220α3-50], where α∈R. Suppose Q=[qij] is a matrix satisfying PQ=kI3 for some non-zero k∈R. If q23=-k8 and |Q|=k22, then α2+k2 is equal to

Q 26 : Consider a matrix A=[αβλα2β2λ2β+λλ+αα+β] where α,β,λ are three distinct natural numbers. If det(adj(adj(adj(adjA))))(α-β)16(β-γ)16(γ-α)16=232×316, then the number of such 3-triples (α,β,γ) is _______ .

Q 27 : Let A=[111–201135]. Then the sum of all elements of the matrix adj(adj(2adjA)–1)) is equal to : [2026]

Q 28 : Let A be a 3×3 matrix such that AT=[101]=[522], AT=[001]=[311], A=[101]=[344] and A=[001]=[131]. If det (A) = 1, then det(adj(A2+A)) is equal to : [2026]

Q 29 : Let A=[α12230045] and B=[1000–5α004α–2α]+adj(A). If det (B) = 66, then det (adj (A)) equals : [2026]

Q 30 : Consider the matrices A=[2–24–2] and B=[3913]. If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

Q 31 : Let A=[–11–1101001] satisfy A2+α(adj(adj(A)))+β(adj(A)(adj(adj(A))))=[2–22–20–100–1] for some α,β∈R. Then (α–β)2 is equal to _______. [2026]

Q 32 : If A=[2294] and I=[1001] then 20A-1 = ____

Q 33 : A is 3×3 matrix and |A|=4 then |adj(adjA)|=4k. Then k is _____

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Q 12 :

If $Δ =$ $| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} |$ $,$ then the cofactor $A_{21}$ is:

Q 13 :

If $Δ =$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$ and $A_{i j}$ is the cofactor of $a_{i j}$ , then the value of $Δ$ is given by:

Q 14 :

If $A = [\begin{matrix} 2 & 3 \\ - 4 & - 6 \end{matrix}],$ then which of the following is true?

Q 15 :

If A and B are invertible matrices, then which of the following is not correct?

Q 16 :

If $A = [\begin{matrix} 2 & λ & - 3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{matrix}],$ then $A^{- 1}$ exists if:

Q 17 :

If $A = [\begin{matrix} 2 & 1 & 0 \\ 3 & 1 & 2 \\ 0 & 4 & - 1 \end{matrix}],$ then $| adj (A) |$ is equal to:

Q 18 :

If $A$ and $B$ are invertible matrices of order 3, $| A |$ $= 2$ and $| {(A B)}^{- 1} |$ $= - \frac{1}{6},$ then the value of $| B |$ is:

Q 19 :

Let $A = [\begin{matrix} 1 & - 2 & 3 \\ 1 & 2 & 1 \\ λ & 2 & - 3 \end{matrix}] .$ If $A^{- 1}$ does not exist, then $λ =$

Q 20 :

The sum of the products of elements of any row with the cofactors of corresponding elements is:

Q 21 :

If A is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ({(2 A)}^{- 1}))))) = 2^{m} 3^{n} .$ Then $m + 2 n$ is equal to:

Q 23 :

If $A$ is a square matrix of order $n$ then, $\underset{(n - 1) times}{\underset{⏟}{(adj adj adj \dots adj A)} \cdot} \underset{n times}{\underset{⏟}{(adj adj adj \dots adj A)}}$ is equal to:

Q 24 :

If $2 \sin α \sin β + 3 \cos β + 5 \cos α \sin β = \sqrt{38} \forall α, β \in R,$ then $\det (adj (adj A))$ is equal to (where $A = [\begin{matrix} \tan β & - \frac{1}{3} \\ 1 & \sin α \end{matrix}]$ ) is equal to ______

Q 27 :

Let $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$ . Then the sum of all elements of the matrix $adj (adj {(2 adj A)}^{- 1}))$ is equal to : [2026]

Q 29 :

Let $A = [\begin{matrix} α & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{matrix}]$ and $B = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 5 α & 0 \\ 0 & 4 α & - 2 α \end{matrix}] + adj (A)$ . If det (B) = 66, then det (adj (A)) equals : [2026]

Q 30 :

Consider the matrices $A = [\begin{matrix} 2 & - 2 \\ 4 & - 2 \end{matrix}]$ and $B = [\begin{matrix} 3 & 9 \\ 1 & 3 \end{matrix}]$ . If matrices P and Q are such that PA = B and AQ = B, then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______. [2026]

Q 32 :

If $A = [\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix}]$ and $I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ then $20 A^{- 1}$ = ____

Q 33 :

A is $3 \times 3$ matrix and $| A |$ $= 4$ then $| a d j (a d j A) |$ $= 4^{k}$ . Then $k$ is _____