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

Q 11 :
Let $A = [a_{i j}]$ be a matrix of order $3 \times 3$ , with $a_{i j} = {(\sqrt{2})}^{i + j}$ . If the sum of all the elements in the third row of $A^{2}$ is $α + β \sqrt{2}, α, β \in Z$ , then $α + β$ is equal to : [2025]

168
224
210
280

1

2

3

4

(2)

We have,

$A = [a_{i j}] of order 3 \times 3 with a_{i j} = {(\sqrt{2})}^{i + j}$

$A = [\begin{matrix} {(\sqrt{2})}^{2} & {(\sqrt{2})}^{3} & {(\sqrt{2})}^{4} \\ {(\sqrt{2})}^{3} & {(\sqrt{2})}^{4} & {(\sqrt{2})}^{5} \\ {(\sqrt{2})}^{4} & {(\sqrt{2})}^{5} & {(\sqrt{2})}^{6} \end{matrix}] = [\begin{matrix} 2 & 2 \sqrt{2} & 4 \\ 2 \sqrt{2} & 4 & 4 \sqrt{2} \\ 4 & 4 \sqrt{2} & 8 \end{matrix}] = 2 [\begin{matrix} 1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2 \sqrt{2} \\ 2 & 2 \sqrt{2} & 4 \end{matrix}]$

$\Rightarrow A^{2} = 4 [\begin{matrix} 1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2 \sqrt{2} \\ 2 & 2 \sqrt{2} & 4 \end{matrix}] [\begin{matrix} 1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2 \sqrt{2} \\ 2 & 2 \sqrt{2} & 4 \end{matrix}] = 4 [\begin{matrix} 7 & 7 \sqrt{2} & 14 \\ 7 \sqrt{2} & 14 & 14 \sqrt{2} \\ 14 & 14 \sqrt{2} & 28 \end{matrix}]$

$∴$ Sum of elements of third row = $4 (14 + 14 \sqrt{2} + 28)$

$= 4 (42 + 14 \sqrt{2}) = 168 + 56 \sqrt{2}$

Comparing the equation $α + β \sqrt{2}$ , we get

$∴ α = 168, β = 56 \Rightarrow α + β = 224$ .

Q 12 :
Let $A = [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}]$ . If for some $θ \in (0, π), A^{2} = A^{T}$ , then the sum of the diagonal elements of the matrix ${(A + I)}^{3} + {(A - I)}^{3} - 6 A$ is equal to __________. [2025]

0%

(6)

We have, $A = [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}]$

Since, A is orthogonal.

$\Rightarrow A A^{T} = A^{T} A = I and A^{T} = A^{- 1}$

Given, $A^{2} = A^{T}$

$\Rightarrow A^{3} = I$

Let $B = {(A + I)}^{3} + {(A - I)}^{3} - 6 A$

$= 2 (A^{3} + 3 A) - 6 A = 2 A^{3} = 2 I$

So, sum of diagonal elements of B = 2(1 + 1 + 1) = 6.

Q 13 :
Let M denote the set of all real matrices of order $3 \times 3$ and let S = {–3, –2, –1, 1, 2}. Let

$S_{1} = {A = [a_{i j}] \in M : A = A^{T} and a_{i j} \in S, \forall i, j}$ ,

$S_{2} = {A = [a_{i j}] \in M : A = - A^{T} and a_{i j} \in S, \forall i, j}$ ,

$S_{3} = {A = [a_{i j}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{i j} \in S, \forall i, j}$ .

If $n (S_{1} \cup S_{2} \cup S_{3}) = 125 α$ , then $α$ equals __________. [2025]

0%

(1613)

Let M denotes the set of all real matrices of order $3 \times 3$ .

$M = {[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]; a_{i j \in R}}$

Now, $S_{1} = {A = [a_{i j}] \in M : A = A^{T} and a_{i j} \in S, \forall i, j}$

$∴$ Number of elements in $S_{1} = 5^{6}$

$S_{2} = {A = [a_{i j}] \in M : A = - A^{T} and a_{i j} \in S, \forall i, j}$

$\Rightarrow$ Number of elements in $S_{2} = 0$

$S_{3} = {A = [a_{i j}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{i j} \in S, \forall i, j}$

$\Rightarrow$ Number of elements in $S_{3} = 12 \times 5^{6}$

[ $∵$ Possible cases are (1, 2, –3) $\to$ 3!, (1, 1, –2) $\to$ 3 and (–1, –1, 2) $\to$ 3]

$∴ n (S_{1} \cap S_{3}) = 12 \times 5^{3}$

Now, $n (S_{1} \cup S_{2} \cup S_{3}) = 5^{6} + 12 \times 5^{6} + 0 - 12 \times 5^{3}$

$= 5^{6} (1 + 12) - 12 \times 5^{3} = 5^{3} [13 \times 5^{3} - 12) = 125 α$

$\Rightarrow α = 1613$ .

Q 14 :
Let $S = {m \in Z : A^{m^{2}} + A^{m} = 3 I - A^{- 6}}$ , where $A = [\begin{matrix} 2 & - 1 \\ 1 & 0 \end{matrix}]$ . Then n(S) is equal to __________. [2025]

0%

(2)

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

$A^{4} = [\begin{matrix} 5 & - 4 \\ 4 & - 3 \end{matrix}]$ and so on $A^{6} = [\begin{matrix} 7 & - 6 \\ 6 & - 5 \end{matrix}]$

$\Rightarrow A^{m} = [\begin{matrix} m + 1 & - m \\ m & - m + 1 \end{matrix}] and A^{m^{2}} = [\begin{matrix} m^{2} + 1 & - m^{2} \\ m^{2} & - (m^{2} - 1) \end{matrix}]$

Now, $A^{m^{2}} + A^{m} = 3 I - A^{- 6}$

$\Rightarrow [\begin{matrix} m^{2} + 1 & - m^{2} \\ m^{2} & - (m^{2} - 1) \end{matrix}] + [\begin{matrix} m + 1 & - m \\ m & - (m - 1) \end{matrix}]$

$= 3 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] - [\begin{matrix} - 5 & 6 \\ - 6 & 7 \end{matrix}] = [\begin{matrix} 8 & - 6 \\ 6 & - 4 \end{matrix}]$

$\Rightarrow m^{2} + 1 + m + 1 \Rightarrow 8 = m^{2} + m - 6 = 0 \Rightarrow m = - 3, 2$

$∴ n (S) = 2$ .

Q 15 :
Let $A = {[a_{i j}]}_{2 \times 2}$ , where $a_{i j} = 0$ for all $i, j$ and $A^{2} = I$ . Let $a$ be the sum of all diagonal elements of $A$ and $b =$ $| A |$ . Then $3 a^{2} + 4 b^{2}$ is equal to [2023]

3
7
4
14

1

2

3

4

(3)

Let
$A = [\begin{matrix} m & n \\ q & p \end{matrix}]$ $∵ A^{2} = I$

$\Rightarrow [\begin{matrix} m^{2} + q n & m n + n p \\ q m + q p & n q + p^{2} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$\Rightarrow m^{2} + q n = 1$ ...(i); $n (m + p) = 0$ ...(ii)

$q (m + p) = 0$ ...(iii), $n q + p^{2} = 1$ ...(iv)

$\Rightarrow m^{2} - p^{2} = 0 \Rightarrow m = \pm p [Using (i) & (iv)]$

Also, $m + p = 0$

Now, let $a = m + p$ and $b = m p - q n$

$So, 3 a^{2} + 4 b^{2} = 3 \times q^{2} + 4 {(m p - q n)}^{2}$

$= 4 {(- m^{2} - q n)}^{2} = 4 \times 1 = 4$ $[Using (i)]$

Q 16 :
Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in ℕ$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P,$ then $α + β + γ - δ$ is equal to [2023]

40
22
18
24

1

2

3

4

(4)

We have,

$p^{α} + p^{β} = γ I - 29 p ...(i)$

$p^{α} - p^{β} = δ I - 13 p ...(ii)$

Now, $p^{2} = I - p$

$p^{3} = p - p^{2} = p - I + p = 2 p - I$

$p^{4} = 2 p^{2} - p = 2 (I - p) - p = 2 I - 3 p$

$p^{5} = 2 p - 3 p^{2} = 2 p - 3 (I - p) = 5 p - 3 I$

$p^{6} = 5 p^{2} - 3 p = 5 (I - p) - 3 p = 5 I - 8 p$

$p^{7} = 5 p - 8 p^{2} = 5 p - 8 (I - p) = 13 p - 8 I$

$p^{8} = 13 p^{2} - 8 p = 13 (I - p) - 8 p = 13 I - 21 p$

$p^{8} + p^{6} = 18 I - 29 p ...(iv)$

$p^{8} - p^{6} = 8 I - 13 p ...(v)$

After comparing (iv) with (i) and (v) with (ii), we get:

$α = 8, β = 6, γ = 18, δ = 8$

$∴ α + β + γ - δ = 8 + 6 + 18 - 8 = 24$

Q 17 :
Let $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}], A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] and Q = P A P^{T} .$ If $P^{T} Q^{2007} P = [\begin{matrix} a & b \\ c & d \end{matrix}],$ then $2 a + b - 3 c - 4 d$ is equal to [2023]

2006
2004
2005
2007

1

2

3

4

(3)

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

$Q = P A P^{T}$

Also, $P^{T} P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{- 1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}] [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{- 1}{2} & \frac{\sqrt{3}}{2} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$Q^{2007} = (P A P^{T}) (P A P^{T}) \dots (2007 times) = P A^{2007} P^{T} [∵ P^{T} P = I]$

$∴ P^{T} Q^{2007} P = P^{T} P A^{2007} P^{T} P = A^{2007}$

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

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

Continuing in the same way, we get:

$A^{2007} = [\begin{matrix} 1 & 2007 \\ 0 & 1 \end{matrix}]$

$∴ P^{T} Q^{2007} P = [\begin{matrix} 1 & 2007 \\ 0 & 1 \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] (Given)$

$\Rightarrow a = 1, b = 2007, c = 0, d = 1$

$∴ 2 a + b - 3 c - 4 d = 2 (1) + 2007 - 3 (0) - 4 (1) = 2005$

Q 18 :
Let $A = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] .$ If $B = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] A [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}],$ then the sum of all the elements of the matrix $\sum_{n = 1}^{50} B^{n}$ is equal to [2023]

50
100
75
125

1

2

3

4

(2)

$A = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}]; B = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] A [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

Let $P = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}], Q = [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$B = P A Q$

$B^{2} = (P A Q) (P A Q) = P A Q P A Q = P A^{2} Q$

As, $Q P = [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$A^{2} = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & \frac{2}{51} \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$A^{3} = [\begin{matrix} 1 & \frac{2}{51} \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & \frac{3}{51} \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

Similarly, $A^{n} = [\begin{matrix} 1 & \frac{n}{51} \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$B^{n} = P A^{n} Q = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] [\begin{matrix} 1 & \frac{n}{51} \\ 0 & 1 \end{matrix}] [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$B^{n} = [\begin{matrix} 1 + \frac{n}{51} & \frac{n}{51} \\ \frac{- n}{51} & 1 - \frac{n}{51} \end{matrix}] \begin{matrix} \end{matrix}$

$\sum_{n = 1}^{50} B^{n} = [\begin{matrix} 50 + 25 & 25 \\ - 25 & 50 - 25 \end{matrix}] = [\begin{matrix} 75 & 25 \\ - 25 & 25 \end{matrix}] \begin{matrix} \end{matrix}$

Sum of the elements = 100

Q 19 :
The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is [2023]

$10^{6}$
$10^{9}$
$6^{10}$
$9^{10}$

1

2

3

4

(1)

Let $A = {[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]}_{3 \times 3}$ be the matrix

A is a symmetric matrix

$∴$ $A = A'$

$\Rightarrow [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] = [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}] \Rightarrow b = d, c = g, f = h$

$∴$ $A = [\begin{matrix} a & b & c \\ b & e & f \\ c & f & i \end{matrix}]$

Each element is chosen from the set {0, 1, 2, 3, ...., 9}

Choice for each element = 10

$∴ Number of symmetric matrices = 10^{6}$

Q 20 :
If $A = \frac{1}{2} [\begin{matrix} 1 & \sqrt{3} \\ - \sqrt{3} & 1 \end{matrix}],$ then [2023]

$A^{30} - A^{25} = 2 I$
$A^{30} = A^{25}$
$A^{30} + A^{25} - A = I$
$A^{30} + A^{25} + A = I$

1

2

3

4

(3)

$A = \frac{1}{2} [\begin{matrix} 1 & \sqrt{3} \\ - \sqrt{3} & 1 \end{matrix}] \Rightarrow A = [\begin{matrix} \cos 60 ° & \sin 60 ° \\ - \sin 60 ° & \cos 60 ° \end{matrix}] \begin{matrix} \end{matrix}$

Let $A = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}], where α = \frac{π}{3}$

$A^{2} = [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}] [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}] \begin{matrix} \end{matrix}$

$= [\begin{matrix} \cos 2 α & \sin 2 α \\ - \sin 2 α & \cos 2 α \end{matrix}] \begin{matrix} \end{matrix}$

$A^{30} = [\begin{matrix} \cos 30 α & \sin 30 α \\ - \sin 30 α & \cos 30 α \end{matrix}]; A^{30} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = I$

$A^{25} = [\begin{matrix} \cos 25 α & \sin 25 α \\ - \sin 25 α & \cos 25 α \end{matrix}] = [\begin{matrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{- \sqrt{3}}{2} & \frac{1}{2} \end{matrix}]$

$\Rightarrow A^{25} = A \Rightarrow A^{25} - A = 0$

Topic Question Set

Q 11 :
Let $A = [a_{i j}]$ be a matrix of order $3 \times 3$ , with $a_{i j} = {(\sqrt{2})}^{i + j}$ . If the sum of all the elements in the third row of $A^{2}$ is $α + β \sqrt{2}, α, β \in Z$ , then $α + β$ is equal to : [2025]

Q 12 :
Let $A = [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}]$ . If for some $θ \in (0, π), A^{2} = A^{T}$ , then the sum of the diagonal elements of the matrix ${(A + I)}^{3} + {(A - I)}^{3} - 6 A$ is equal to __________. [2025]

0%

0%

Q 14 :
Let $S = {m \in Z : A^{m^{2}} + A^{m} = 3 I - A^{- 6}}$ , where $A = [\begin{matrix} 2 & - 1 \\ 1 & 0 \end{matrix}]$ . Then n(S) is equal to __________. [2025]

0%

Q 15 :
Let $A = {[a_{i j}]}_{2 \times 2}$ , where $a_{i j} = 0$ for all $i, j$ and $A^{2} = I$ . Let $a$ be the sum of all diagonal elements of $A$ and $b =$ $| A |$ . Then $3 a^{2} + 4 b^{2}$ is equal to [2023]

Q 16 :
Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in ℕ$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P,$ then $α + β + γ - δ$ is equal to [2023]

Q 18 :
Let $A = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] .$ If $B = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] A [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}],$ then the sum of all the elements of the matrix $\sum_{n = 1}^{50} B^{n}$ is equal to [2023]

Q 19 :
The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is [2023]

Q 20 :
If $A = \frac{1}{2} [\begin{matrix} 1 & \sqrt{3} \\ - \sqrt{3} & 1 \end{matrix}],$ then [2023]

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Topic Question Set

Q 11 : Let A=[aij] be a matrix of order 3×3, with aij=(2)i+j. If the sum of all the elements in the third row of A2 is α+β2, α, β∈Z, then α+β is equal to : [2025]

Q 12 : Let A=[cosθ0–sinθ010sinθ0cosθ]. If for some θ∈(0,π), A2=AT, then the sum of the diagonal elements of the matrix (A+I)3+(A–I)3–6A is equal to __________. [2025]

0%

0%

Q 14 : Let S={m∈Z : Am2+Am=3I–A–6}, where A=[2–110]. Then n(S) is equal to __________. [2025]

0%

Q 15 : Let A=[aij]2×2, where aij=0 for all i,j and A2=I. Let a be the sum of all diagonal elements of A and b=|A|. Then 3a2+4b2 is equal to [2023]

Q 16 : Let P be a square matrix such that P2=I-P. For α,β,γ,δ∈ℕ, if Pα+Pβ=γI-29P and Pα-Pβ=δI-13P, then α+β+γ-δ is equal to [2023]

Q 17 : Let P= [3212-1232], A=[1101] and Q=PAPT. If PTQ2007P=[abcd], then 2a+b-3c-4d is equal to [2023]

Q 18 : Let A=[115101]. If B=[12-1-1]A[-1-211], then the sum of all the elements of the matrix ∑n=150Bn is equal to [2023]

Q 19 : The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is [2023]

Q 20 : If A=12[13-31], then [2023]

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Q 11 :
Let $A = [a_{i j}]$ be a matrix of order $3 \times 3$ , with $a_{i j} = {(\sqrt{2})}^{i + j}$ . If the sum of all the elements in the third row of $A^{2}$ is $α + β \sqrt{2}, α, β \in Z$ , then $α + β$ is equal to : [2025]

Q 12 :
Let $A = [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}]$ . If for some $θ \in (0, π), A^{2} = A^{T}$ , then the sum of the diagonal elements of the matrix ${(A + I)}^{3} + {(A - I)}^{3} - 6 A$ is equal to __________. [2025]

Q 14 :
Let $S = {m \in Z : A^{m^{2}} + A^{m} = 3 I - A^{- 6}}$ , where $A = [\begin{matrix} 2 & - 1 \\ 1 & 0 \end{matrix}]$ . Then n(S) is equal to __________. [2025]

Q 15 :
Let $A = {[a_{i j}]}_{2 \times 2}$ , where $a_{i j} = 0$ for all $i, j$ and $A^{2} = I$ . Let $a$ be the sum of all diagonal elements of $A$ and $b =$ $| A |$ . Then $3 a^{2} + 4 b^{2}$ is equal to [2023]

Q 16 :
Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in ℕ$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P,$ then $α + β + γ - δ$ is equal to [2023]

Q 18 :
Let $A = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] .$ If $B = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] A [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}],$ then the sum of all the elements of the matrix $\sum_{n = 1}^{50} B^{n}$ is equal to [2023]

Q 19 :
The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is [2023]

Q 20 :
If $A = \frac{1}{2} [\begin{matrix} 1 & \sqrt{3} \\ - \sqrt{3} & 1 \end{matrix}],$ then [2023]