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

Q 1 :
For $α, β \in R$ and a natural number $n$ , let $A_{r} =$ $| \begin{matrix} r & 1 & \frac{n^{2}}{2} + α \\ 2 r & 2 & n^{2} - β \\ 3 r - 2 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$ . Then $2 A_{10} - A_{8}$ is [2024]

$0$
$4 α + 2 β$
$2 n$
$2 α + 4 β$

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3

4

(B)

We have, $2 A_{10} - A_{8} =$ $| \begin{matrix} 20 & 1 & \frac{n^{2}}{2} + α \\ 40 & 2 & n^{2} - β \\ 56 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$ $-$ $| \begin{matrix} 8 & 1 & \frac{n^{2}}{2} + α \\ 16 & 2 & n^{2} - β \\ 22 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$

$=$ $| \begin{matrix} 20 - 8 & 1 & \frac{n^{2}}{2} + α \\ 40 - 16 & 2 & n^{2} - β \\ 56 - 22 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$ $=$ $| \begin{matrix} 12 & 1 & \frac{n^{2}}{2} + α \\ 24 & 2 & n^{2} - β \\ 34 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$

$=$ $| \begin{matrix} 0 & 1 & \frac{n^{2}}{2} + α \\ 0 & 2 & n^{2} - β \\ - 2 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$ $(Applying C_{1} \to C_{1} - 12 C_{2})$

$= - 2 (n^{2} - β - n^{2} - 2 α)$

$= - 2 (- β - 2 α) = 4 α + 2 β$

Q 2 :
Let $A = [\begin{matrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{matrix}] .$ If $A^{3} = 4 A^{2} - A - 21 I,$ where $I$ is the identity matrix of order $3 \times 3$ , then $2 a + 3 b$ is equal to [2024]

$- 12$
$- 13$
$- 10$
$- 9$

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2

3

4

(B)

$A = [\begin{matrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{matrix}],$ $A^{3} = 4 A^{2} - A - 21 I$

characteristic equation of $A$ is given by

$x^{3} - 4 x^{2} + x + 21 = 0$

Sum of roots = 4 = trace of A

$\Rightarrow 2 + 3 + b = 4 \Rightarrow b = - 1$

Also, product of roots = $- 21$ = det A

$\Rightarrow 2 (3 b - 5) - a (b) = - 21 \Rightarrow - 6 - 10 + a = - 21$

$\Rightarrow a = - 21 + 16 = - 5$ $∴$ $2 a + 3 b = - 10 - 3 = - 13$

Q 3 :
If $α \neq a,$ $β \neq b,$ $γ \neq c$ and $| \begin{matrix} α & b & c \\ a & β & c \\ a & b & γ \end{matrix} |$ $= 0,$ then $\frac{a}{α - a} + \frac{b}{β - b} + \frac{γ}{γ - c}$ is equal to: [2024]

1
2
0
3

1

2

3

4

(C)

$| \begin{matrix} α & b & c \\ a & β & c \\ a & b & γ \end{matrix} |$ $= 0$

$R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}$

$\Rightarrow$ $| \begin{matrix} α - a & b - β & 0 \\ 0 & β - b & c - γ \\ a & b & γ \end{matrix} |$ $= 0$

Taking out $α - a, β - b$ and $γ - c$ common from column-1, 2 and 3 respectively,

$\Rightarrow (α - a) (β - b) (γ - c)$ $| \begin{matrix} 1 & - 1 & 0 \\ 0 & 1 & - 1 \\ \frac{a}{α - a} & \frac{b}{β - b} & \frac{γ}{γ - c} \end{matrix} |$ $= 0$

$\Rightarrow (α - a) (β - b) (γ - c) [1 (\frac{γ}{γ - c} + \frac{b}{β - b}) + 1 (0 + \frac{a}{α - a})] = 0$

$\Rightarrow (α - a) (β - b) (γ - c) [\frac{γ}{γ - c} + \frac{b}{β - b} + \frac{a}{α - a}] = 0$

$\Rightarrow [\frac{γ}{γ - c} + \frac{b}{β - b} + \frac{a}{α - a}] = 0$ $(∵ α \neq a, β \neq b, γ \neq c)$

$∴$ $\frac{γ}{γ - c} + \frac{b}{β - b} + \frac{a}{α - a} = 0$

Q 4 :
If $A = [\begin{matrix} \sqrt{2} & 1 \\ - 1 & \sqrt{2} \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}], C = A B A^{T}$ and $X = A^{T} C^{2} A,$ then det $X$ is equal to [2024]

729
243
891
27

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2

3

4

(A)

Given, $X = A^{T} C^{2} A$ and $C = A B A^{T}$

$\Rightarrow | X | = | A^{T} C^{2} A | \Rightarrow | X | = | A | | C |^{2} | A^{T} |$

$\Rightarrow | C | = | A B A^{T} | = | A | | B | | A^{T} |$

As $| A | = 3, | A^{T} | = 3, | B | = 1, | C | = 3 \times 1 \times 3 = 9$

$∴$ $| X | = 3 \times 9^{2} \times 3 = 9 \times 81 = 729$

Q 5 :
The values of $α,$ for which $| \begin{matrix} 1 & \frac{3}{2} & α + \frac{3}{2} \\ 1 & \frac{1}{3} & α + \frac{1}{3} \\ 2 α + 3 & 3 α + 1 & 0 \end{matrix} |$ $= 0,$ lie in the interval [2024]

$(- \frac{3}{2}, \frac{3}{2})$
$(- 2, 1)$
$(- 3, 0)$
$(0, 3)$

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2

3

4

(C)

Given, $| \begin{matrix} 1 & \frac{3}{2} & α + \frac{3}{2} \\ 1 & \frac{1}{3} & α + \frac{1}{3} \\ 2 α + 3 & 3 α + 1 & 0 \end{matrix} |$ $= 0$

$\Rightarrow (2 α + 3) (\frac{7 α}{6}) - (3 α + 1) (\frac{- 7}{6}) = 0$

$\Rightarrow 14 α^{2} + 42 α + 7 = 0 \Rightarrow 2 α^{2} + 6 α + 1 = 0$

$Roots of this equation are α = \frac{- 3 \pm \sqrt{7}}{2}$

$\frac{- 3 + \sqrt{7}}{2} \in (- 3, 0) and \frac{- 3 - \sqrt{7}}{2} \in (- 3, 0)$

Q 6 :
Let $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & α & β \\ 0 & β & α \end{matrix}]$ and $| 2 A |^{3} = 2^{21}$ where $α, β \in Z .$ Then a value of $α$ is [2024]

5
9
17
3

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3

4

(A)

Given, $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & α & β \\ 0 & β & α \end{matrix}]$

$\Rightarrow | A | = α^{2} - β^{2}$ ...(i)

Also, $| 2 A |^{3} = 2^{21}$

$\Rightarrow {(2^{3} | A |)}^{3} = 2^{21} \Rightarrow | A |^{3} = 2^{12} \Rightarrow | A | = 2^{4}$

From (i), we get, $α^{2} - β^{2} = 2^{4}$

$\Rightarrow α^{2} - β^{2} = 16 \Rightarrow (α + β) (α - β) = 8 \times 2$

So, $α + β = 8 and α - β = 2$

Solving, we get, $α = 5$ and $β = 3$

Q 7 :
Let $A = [\begin{matrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{matrix}]$ and $P = [\begin{matrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{matrix}] .$ The sum of the prime factors of $| P^{- 1} A P - 2 I |$ is equal to [2024]

66
27
26
23

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2

3

4

(C)

$P^{- 1} A P - 2 I = P^{- 1} (A - 2 I) P$

$\Rightarrow | P^{- 1} A P - 2 I | = | P^{- 1} | | A - 2 I | | P |$

$= | A - 2 I |$ $(∵ | P^{- 1} | \cdot | P | = 1)$

$∴$ $\det (A - 2 I) =$ $| \begin{matrix} 0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0 \end{matrix} |$

$= - 1 (- 33) + 2 (18) = 69 = 3 \times 23$

Sum of prime factors = 23 + 3 = 26

Q 8 :
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 3 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] .$ Then the maximum value of $d e t (A)$ is _______. [2024]

0%

(27)

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

Now, $A [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 3 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] \Rightarrow [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 3 \end{matrix}]$

$a_{11} + a_{12} + a_{13} = 3$

$a_{21} + a_{22} + a_{23} = 3$

$a_{31} + a_{32} + a_{33} = 3$

Now, for maximum value of $d e t (A) = a_{i j} = {\begin{matrix} 0, & i \neq j \\ 3, & i = j \end{matrix}$

$∴$ $| A | = 27$

Q 9 :
Let $A = [\begin{matrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}],$ $B = [\begin{matrix} B_{1}, & B_{2}, & B_{3} \end{matrix}]$ , where $B_{1}, B_{2}, B_{3}$ are column matrices, and $A B_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], A B_{2} = [\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}], A B_{3} = [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}] .$ If $α = | B |$ and $β$ is the sum of all the diagonal elements of B, then $α^{3} + β^{3}$ is equal to _____ . [2024]

0%

(28)

Let $B_{1} = [\begin{matrix} a_{1} \\ b_{1} \\ c_{1} \end{matrix}],$ $A B_{1} = [\begin{matrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}] [\begin{matrix} a_{1} \\ b_{1} \\ c_{1} \end{matrix}] = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$

$\Rightarrow 2 a_{1} + c_{1} = 1, a_{1} + b_{1} = 0, a_{1} + c_{1} = 0$

On solving, we get $a_{1} = 1, b_{1} = - 1, c_{1} = - 1$ ...(i)

Similarly, let $B_{2} = [\begin{matrix} a_{2} \\ b_{2} \\ c_{2} \end{matrix}], A B_{2} = [\begin{matrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}] [\begin{matrix} a_{2} \\ b_{2} \\ c_{2} \end{matrix}] = [\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}]$

$\Rightarrow 2 a_{2} + c_{2} = 2, a_{2} + b_{2} = 3$ and $a_{2} + c_{2} = 0$

On solving, we get $a_{2} = 2, b_{2} = 1$ and $c_{2} = - 2$ ...(ii)

Similarly, let $B_{3} = [\begin{matrix} a_{3} \\ b_{3} \\ c_{3} \end{matrix}], A B_{3} = [\begin{matrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}] [\begin{matrix} a_{3} \\ b_{3} \\ c_{3} \end{matrix}] = [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]$

$\Rightarrow 2 a_{3} + c_{3} = 3, a_{3} + b_{3} = 2$ and $a_{3} + c_{3} = 1$

$\Rightarrow a_{3} = 2, b_{3} = 0, c_{3} = - 1$ ...(iii)

Thus, $B = [\begin{matrix} 1 & 2 & 2 \\ - 1 & 1 & 0 \\ - 1 & - 2 & - 1 \end{matrix}]$ [By using (i), (ii) & (iii)]

$α = | B | = 1 (- 1 + 0) - 2 (1 - 0) + 2 (2 + 1) = - 1 - 2 + 6 = 3$

and $β =$ sum of diagonal elements of $B = 1 + 1 + (- 1) = 1$

Hence, $α^{3} + β^{3} = {(3)}^{3} + {(1)}^{3} = 28$

Q 10 :
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $| A - x I | = 0$ be $- 1$ and $3$ , then the sum of the diagonal elements of the matrix $A^{2}$ is _______. [2024]

0%

(10)

Let $A = {[\begin{matrix} a & b \\ c & d \end{matrix}]}_{2 \times 2}, I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Roots are $- 1, 3$

Sum of roots $= - 1 + 3 = 2 = t r (A)$

$a + d = 2$ ...(i)

Product of roots $= (- 1) \times 3 = - 3 = \det (A)$

$a d - b c = - 3$ ...(ii)

$A^{2} = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a^{2} + b c & a b + b d \\ a c + d c & b c + d^{2} \end{matrix}]$

$tr (A^{2}) = a^{2} + 2 b c + d^{2} = {(a + d)}^{2} - 2 a d + 2 b c$

$= 2^{2} - 2 (- 3) = 4 + 6 = 10$

Topic Question Set

Q 1 :
For $α, β \in R$ and a natural number $n$ , let $A_{r} =$ $| \begin{matrix} r & 1 & \frac{n^{2}}{2} + α \\ 2 r & 2 & n^{2} - β \\ 3 r - 2 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$ . Then $2 A_{10} - A_{8}$ is [2024]

Q 2 :
Let $A = [\begin{matrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{matrix}] .$ If $A^{3} = 4 A^{2} - A - 21 I,$ where $I$ is the identity matrix of order $3 \times 3$ , then $2 a + 3 b$ is equal to [2024]

Q 3 :
If $α \neq a,$ $β \neq b,$ $γ \neq c$ and $| \begin{matrix} α & b & c \\ a & β & c \\ a & b & γ \end{matrix} |$ $= 0,$ then $\frac{a}{α - a} + \frac{b}{β - b} + \frac{γ}{γ - c}$ is equal to: [2024]

Q 4 :
If $A = [\begin{matrix} \sqrt{2} & 1 \\ - 1 & \sqrt{2} \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}], C = A B A^{T}$ and $X = A^{T} C^{2} A,$ then det $X$ is equal to [2024]

Q 5 :
The values of $α,$ for which $| \begin{matrix} 1 & \frac{3}{2} & α + \frac{3}{2} \\ 1 & \frac{1}{3} & α + \frac{1}{3} \\ 2 α + 3 & 3 α + 1 & 0 \end{matrix} |$ $= 0,$ lie in the interval [2024]

Q 6 :
Let $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & α & β \\ 0 & β & α \end{matrix}]$ and $| 2 A |^{3} = 2^{21}$ where $α, β \in Z .$ Then a value of $α$ is [2024]

Q 7 :
Let $A = [\begin{matrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{matrix}]$ and $P = [\begin{matrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{matrix}] .$ The sum of the prime factors of $| P^{- 1} A P - 2 I |$ is equal to [2024]

Q 8 :
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 3 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] .$ Then the maximum value of $d e t (A)$ is _______. [2024]

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0%

Q 10 :
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $| A - x I | = 0$ be $- 1$ and $3$ , then the sum of the diagonal elements of the matrix $A^{2}$ is _______. [2024]

0%

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Topic Question Set

Q 1 : For α,β∈R and a natural number n, let Ar=|r1n22+α2r2n2-β3r-23n(3n-1)2|. Then 2A10-A8 is [2024]

Q 2 : Let A=[2a013105b]. If A3=4A2-A-21I, where I is the identity matrix of order 3×3, then 2a+3b is equal to [2024]

Q 3 : If α≠a, β≠b, γ≠c and |αbcaβcabγ|=0, then aα-a+bβ-b+γγ-c is equal to: [2024]

Q 4 : If A=[21-12], B=[1011], C=ABATand X=ATC2A, then det X is equal to [2024]

Q 5 : The values of α, for which |132α+32113α+132α+33α+10|=0, lie in the interval [2024]

Q 6 : Let A=[1000αβ0βα] and |2A|3=221 where α,β∈Z. Then a value of α is [2024]

Q 7 : Let A=[2126211332] and P=[120502715]. The sum of the prime factors of|P-1AP-2I| is equal to [2024]

Q 8 : Let A be a 3×3 matrix of non-negative real elements such that A[111]=3[111]. Then the maximum value of det(A) is _______. [2024]

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Q 9 : Let A=[201110101], B=[B1,B2,B3], where B1,B2,B3 are column matrices, and AB1=[100], AB2=[230], AB3=[321]. If α=|B| and β is the sum of all the diagonal elements of B, then α3+β3 is equal to _____ . [2024]

0%

Q 10 : Let A be a 2×2 real matrix and I be the identity matrix of order 2. If the roots of the equation |A-xI|=0 be -1 and 3, then the sum of the diagonal elements of the matrix A2 is _______. [2024]

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Q 1 :
For $α, β \in R$ and a natural number $n$ , let $A_{r} =$ $| \begin{matrix} r & 1 & \frac{n^{2}}{2} + α \\ 2 r & 2 & n^{2} - β \\ 3 r - 2 & 3 & \frac{n (3 n - 1)}{2} \end{matrix} |$ . Then $2 A_{10} - A_{8}$ is [2024]

Q 2 :
Let $A = [\begin{matrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{matrix}] .$ If $A^{3} = 4 A^{2} - A - 21 I,$ where $I$ is the identity matrix of order $3 \times 3$ , then $2 a + 3 b$ is equal to [2024]

Q 3 :
If $α \neq a,$ $β \neq b,$ $γ \neq c$ and $| \begin{matrix} α & b & c \\ a & β & c \\ a & b & γ \end{matrix} |$ $= 0,$ then $\frac{a}{α - a} + \frac{b}{β - b} + \frac{γ}{γ - c}$ is equal to: [2024]

Q 4 :
If $A = [\begin{matrix} \sqrt{2} & 1 \\ - 1 & \sqrt{2} \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}], C = A B A^{T}$ and $X = A^{T} C^{2} A,$ then det $X$ is equal to [2024]

Q 5 :
The values of $α,$ for which $| \begin{matrix} 1 & \frac{3}{2} & α + \frac{3}{2} \\ 1 & \frac{1}{3} & α + \frac{1}{3} \\ 2 α + 3 & 3 α + 1 & 0 \end{matrix} |$ $= 0,$ lie in the interval [2024]

Q 6 :
Let $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & α & β \\ 0 & β & α \end{matrix}]$ and $| 2 A |^{3} = 2^{21}$ where $α, β \in Z .$ Then a value of $α$ is [2024]

Q 7 :
Let $A = [\begin{matrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{matrix}]$ and $P = [\begin{matrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{matrix}] .$ The sum of the prime factors of $| P^{- 1} A P - 2 I |$ is equal to [2024]

Q 8 :
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 3 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] .$ Then the maximum value of $d e t (A)$ is _______. [2024]

Q 10 :
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $| A - x I | = 0$ be $- 1$ and $3$ , then the sum of the diagonal elements of the matrix $A^{2}$ is _______. [2024]