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

Q 21 :

$Let D_{k} =$ $| \begin{matrix} 1 & 2 k & 2 k - 1 \\ n & n^{2} + n + 2 & n^{2} \\ n & n^{2} + n & n^{2} + n + 2 \end{matrix} |$ . $If \sum_{k = 1}^{n} D_{k} = 96, then n is equal to$ _________ . [2023]

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

$∴$ $\sum_{k = 1}^{n} D_{k} =$ $| \begin{matrix} \sum_{k = 1}^{n} 1 & \sum_{k = 1}^{n} 2 k & \sum_{k = 1}^{n} (2 k - 1) \\ n & n^{2} + n + 2 & n^{2} \\ n & n^{2} + n & n^{2} + n + 2 \end{matrix} |$

$Apply \sum_{k = 1}^{n} 2 k = \frac{2 n (n + 1)}{2} = n (n + 1)$

$\sum_{k = 1}^{n} (2 k - 1) = \sum_{k = 1}^{n} 2 k - \sum_{k = 1}^{n} 1 = \frac{2 n (n + 1)}{2} - n = n^{2} + n - n = n^{2}$

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

$Apply R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - R_{1}$

$\Rightarrow$ $| \begin{matrix} n & n^{2} + n & n^{2} \\ 0 & 2 & 0 \\ 0 & 0 & n + 2 \end{matrix} |$ $= 96$

$\Rightarrow n (2 (n + 2) - 0) = 96 \Rightarrow n (n + 2) = 48$

$\Rightarrow n^{2} + 2 n - 48 = 0 \Rightarrow n^{2} + 8 n - 6 n - 48 = 0 \Rightarrow n (n + 8) - 6 (n + 8) = 0 \Rightarrow (n - 6) (n + 8) = 0$

$\Rightarrow n = 6, - 8$

$Since n cannot be negative, ∴$ $n = 6$

Q 22 :

Among the statements:

$I: If$ $| \begin{matrix} 1 & \cos α & \cos β \\ \cos α & 1 & \cos γ \\ \cos β & \cos γ & 1 \end{matrix} |$ $=$ $| \begin{matrix} 0 & \cos α & \cos β \\ \cos α & 0 & \cos γ \\ \cos β & \cos γ & 0 \end{matrix} |$ $, then \cos^{2} α + \cos^{2} β + \cos^{2} γ = \frac{3}{2},$ and

$II: If$ $| \begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{matrix} |$ $= p x + q, then p^{2} = 196 q^{2} .$ [2026]

only I is true
both are false
only II is true
both are true

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Q 23 :

Let $| A |$ $= 6$ , where A is a $3 \times 3 matrix.$ If $| adj (3 adj (A^{2} \cdot adj (2 A))) |$ $= 2^{m} \cdot 3^{n}, m, n \in N,$ then $m + n$ is equal to _________ . [2026]

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Q 24 :

For some $α, β \in ℝ$ , let $A = [\begin{matrix} α & 2 \\ 1 & 2 \end{matrix}] and B = [\begin{matrix} 1 & 1 \\ 1 & β \end{matrix}]$ be such that $A^{2} - 4 A + 2 I = B^{2} - 3 B + I = O .$ Then $(\det (adj {(A^{3} - B^{3})))}^{2} is equal to$ ________ [2026]

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(225)

$Tr (A) = 4 \Rightarrow α + 2 = 4 \Rightarrow α = 2$

$Tr (B) = 3 \Rightarrow β + 1 = 3 \Rightarrow β = 2$

$A^{2} - 4 A + 2 I = 0$

$A^{3} = 4 A^{2} - 2 A = 16 A - 8 I - 2 A = 14 A - 8 I$

$= [\begin{matrix} 28 & 28 \\ 14 & 28 \end{matrix}] + [\begin{matrix} - 8 & 0 \\ 0 & - 8 \end{matrix}]$

$= A^{3} = [\begin{matrix} 20 & 28 \\ 14 & 20 \end{matrix}]$

$B^{2} - 3 B + I = 0$

$B^{2} = 3 B - I$

$\Rightarrow B^{3} = 3 B^{2} - B = 3 (3 B - I) - B = 8 B - 3 I$

$B^{3} = [\begin{matrix} 8 & 8 \\ 8 & 16 \end{matrix}] + [\begin{matrix} - 3 & 0 \\ 0 & - 3 \end{matrix}] = [\begin{matrix} 5 & 8 \\ 8 & 13 \end{matrix}]$

$A^{3} - B^{3} = [\begin{matrix} 20 & 28 \\ 14 & 20 \end{matrix}] - [\begin{matrix} 5 & 8 \\ 8 & 13 \end{matrix}] = [\begin{matrix} 15 & 20 \\ 6 & 7 \end{matrix}]$

$\Rightarrow$ $| A^{3} - B^{3} |$ $= 105 - 120 = - 15$

$\Rightarrow$ $| adj (A^{3} - B^{3}) |$ $=$ $| A^{3} - B^{3} |$ $= - 15$

$\Rightarrow | adj (A^{3} - B^{3}) |^{2} = 225$

Q 25 :

If $A = [\begin{matrix} 2 & 3 \\ 3 & 5 \end{matrix}],$ then the determinant of the matrix $(A^{2025} - 3 A^{2024} + A^{2023})$ is [2026]

16
12
24
28

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Q 26 :

Let $P = [p_{i j}]$ and $Q = [q_{i j}]$ be two square matrices of order 3 such that $q_{i j} = 2^{(i + j - 1)} p_{i j}$ and $\det (Q) = 2^{10} .$ Then the value of $\det (adj (adj P))$ is: [2026]

124
16
81
32

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Q 27 :

The system of linear equations

$\begin{matrix} x + y + z = 6 \\ 2 x + 5 y + a z = 36 \\ x + 2 y + 3 z = b \end{matrix}$ has [2026]

unique solution for a=8 and b=14
infinitely many solutions for a=8 and b=16
infinitely many solutions for a=8 and b=14
unique solution for a=8 and b=16

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Q 28 :

If the system of equations

$\begin{matrix} 3 x + y + 4 z = 3 \\ 2 x + α y - z = - 3 \\ x + 2 y + z = 4 \end{matrix}$

has no solution, then the value of $α$ is equal to : [2026]

19
23
13
4

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3

4

(1)

Q 29 :

If $X = [[\begin{matrix} x \\ y \\ z \end{matrix}]]$ is a solution of the system of equations AX=B, where $A = [[\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{matrix}]] a n d B = [[\begin{matrix} 4 \\ 0 \\ 2 \end{matrix}]]$ then $| x + y + z |$ is equal to: [2026]

3/2
2
1
3

1

2

3

4

(2)

Topic Question Set

Q 21 :

$Let D_{k} =$ $| \begin{matrix} 1 & 2 k & 2 k - 1 \\ n & n^{2} + n + 2 & n^{2} \\ n & n^{2} + n & n^{2} + n + 2 \end{matrix} |$ . $If \sum_{k = 1}^{n} D_{k} = 96, then n is equal to$ _________ . [2023]

0%

Q 23 :

Let $| A |$ $= 6$ , where A is a $3 \times 3 matrix.$ If $| adj (3 adj (A^{2} \cdot adj (2 A))) |$ $= 2^{m} \cdot 3^{n}, m, n \in N,$ then $m + n$ is equal to _________ . [2026]

0%

Q 24 :

For some $α, β \in ℝ$ , let $A = [\begin{matrix} α & 2 \\ 1 & 2 \end{matrix}] and B = [\begin{matrix} 1 & 1 \\ 1 & β \end{matrix}]$ be such that $A^{2} - 4 A + 2 I = B^{2} - 3 B + I = O .$ Then $(\det (adj {(A^{3} - B^{3})))}^{2} is equal to$ ________ [2026]

0%

Q 25 :

If $A = [\begin{matrix} 2 & 3 \\ 3 & 5 \end{matrix}],$ then the determinant of the matrix $(A^{2025} - 3 A^{2024} + A^{2023})$ is [2026]

Q 26 :

Let $P = [p_{i j}]$ and $Q = [q_{i j}]$ be two square matrices of order 3 such that $q_{i j} = 2^{(i + j - 1)} p_{i j}$ and $\det (Q) = 2^{10} .$ Then the value of $\det (adj (adj P))$ is: [2026]

Q 27 :

The system of linear equations

$\begin{matrix} x + y + z = 6 \\ 2 x + 5 y + a z = 36 \\ x + 2 y + 3 z = b \end{matrix}$ has [2026]

Q 28 :

If the system of equations

$\begin{matrix} 3 x + y + 4 z = 3 \\ 2 x + α y - z = - 3 \\ x + 2 y + z = 4 \end{matrix}$

has no solution, then the value of $α$ is equal to : [2026]

Q 29 :

If $X = [[\begin{matrix} x \\ y \\ z \end{matrix}]]$ is a solution of the system of equations AX=B, where $A = [[\begin{matrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{matrix}]] a n d B = [[\begin{matrix} 4 \\ 0 \\ 2 \end{matrix}]]$ then $| x + y + z |$ is equal to: [2026]

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