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:

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

1

2

3

4

(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:

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

1

2

3

4

(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

1

2

3

4

(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?

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

1

2

3

4

(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

1

2

3

4

(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

1

2

3

4

(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:

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

1

2

3

4

(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 $λ =$

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

1

2

3

4

(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

1

2

3

4

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

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:

Want Us to Email you About Special Offers & Updates?

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:

Want Us to Email you About Special Offers & Updates?

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: