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,

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

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

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

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

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

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

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

$∴ n (S) = 2$ .

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]

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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]

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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=[cos θ0–sin θ010sin θ0cos θ]. 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]

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

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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]