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

Q 21 :
If A and B are two non-zero $n \times n$ matrices such that $A^{2} + B = A^{2} B,$ then [2023]

$A^{2} = I$ or $B = I$
$A^{2} B = B A^{2}$
$A B = I$
$A^{2} B = I$

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(2)

$A^{2} + B = A^{2} B \Rightarrow A^{2} + B - A^{2} B = 0$

$\Rightarrow A^{2} - A^{2} B + B = 0 \Rightarrow A^{2} (I - B) + B = 0$

$\Rightarrow A^{2} (I - B) + B - I = - I \Rightarrow (A^{2} - I) (I - B) = - I$

$\Rightarrow (A^{2} - I) (B - I) = I \Rightarrow A^{2} B - A^{2} - B = 0$

$\Rightarrow A^{2} (B - I) = B \Rightarrow A^{2} = B {(B - I)}^{- 1} \Rightarrow A^{2} = B (A^{2} - I)$

$\Rightarrow A^{2} = B A^{2} - B \Rightarrow A^{2} + B = B A^{2} \Rightarrow A^{2} B = B A^{2}$

Q 22 :
Let $A = [\begin{matrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{- 3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{matrix}] and B = [\begin{matrix} 1 & - i \\ 0 & 1 \end{matrix}],$ where $i = \sqrt{- 1}$ . If $M = A^{T} B A$ , then the inverse of the matrix $A M^{2023} A^{T}$ is [2023]

$[\begin{matrix} 1 & - 2023 i \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} 1 & 0 \\ 2023 i & 1 \end{matrix}]$
$[\begin{matrix} 1 & 2023 i \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} 1 & 0 \\ - 2023 i & 1 \end{matrix}]$

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(3)

Q 23 :
Let A, B, C be $3 \times 3$ matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements:

$(S1) A^{13} B^{26} - B^{26} A^{13} is symmetric$

$(S2) A^{26} C^{13} - C^{13} A^{26} is symmetric$

Then, [2023]

Only S1 is true
Both S1 and S2 are false
Both S1 and S2 are true
Only S2 is true

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(4)

$Since A is symmetric. ∴ A' = A$

$and B and C are skew symmetric.$

$∴ B' = - B and C' = - C$

$S 1 : (A^{13} B^{26} - B^{26} A^{13})' = (A^{13} B^{26})' - (B^{26} A^{13})'$

$= {(B')}^{26} {(A')}^{13} - {(A')}^{13} {(B')}^{26} (∵ (X' Y') = (Y' X'))$

$= {(- B)}^{26} {(A)}^{13} - {(A)}^{13} {(- B)}^{26}$

$= B^{26} A^{13} - A^{13} B^{26} = - (A^{13} B^{26} - B^{26} A^{13})$

Which is skew symmetric.

$∴ S 1 is false.$

$S 2 : (A^{26} C^{13} - C^{13} A^{26})' = (A^{26} C^{13})' - (C^{13} A^{26})'$

$= {(C')}^{13} {(A')}^{26} - {(A')}^{26} {(C')}^{13}$

$= {(- C')}^{13} A^{26} - A^{26} {(- C)}^{13} = - C^{13} A^{26} + A^{26} C^{13}$

Which is symmetric.

$∴ S 2 is true.$

Q 24 :
Let $α$ and $β$ be real numbers. Consider a $3 \times 3$ matrix A such that $A^{2} = 3 A + α I .$ If $A^{4} = 21 A + β I,$ then [2023]

$β = - 8$
$β = 8$
$α = 4$
$α = 1$

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(1)

$Given, A^{2} = 3 A + α I and A^{4} = 21 A + β I$

$Now, A^{2} A^{2} = (3 A + α I) (3 A + α I)$

$= 9 A^{2} + 6 α I A + α^{2} I = 9 (3 A + α I) + 6 α I A + α^{2} I$

$= 27 A + 9 α I + 6 α A + α^{2} I = (27 + 6 α) A + (9 α + α^{2}) I$

$So, 27 + 6 α = 21 \Rightarrow α = - 1$

$\Rightarrow 9 α + α^{2} = β \Rightarrow 9 (- 1) + {(- 1)}^{2} = β \Rightarrow β = - 8$

Q 25 :
Let $A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}) .$ Then the sum of the diagonal elements of the matrix ${(A + I)}^{11}$ is equal to:

2050
4094
6144
4097

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(4)

We have, $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}] \begin{matrix} \end{matrix}$

$A^{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}] = A$

$Similarly, A^{n} = A; n \in N$

$Now, {(A + I)}^{11} = {}^{11}C_{0} A^{11} + {}^{11}C_{1} A^{10} I^{1} + {}^{11}C_{2} A^{9} I^{2} + \dots + {}^{11}C_{11} A^{0} I^{11}$

$= A ({}^{11}C_{0} + {}^{11}C_{1} + \dots + {}^{11}C_{10}) + I [∵ A^{n} = A and I^{n} = I]$

$= A (2^{11} - 1) + I$

$Trace of {(A + I)}^{11} = 1 (2^{11} - 1) + 1 + 4 (2^{11} - 1) + 1 + (- 3) (2^{11} - 1) + 1$

$= 2^{11} + 4 (2^{11}) - 3 - 3 (2^{11}) + 4 = 2 \times 2^{11} + 1$

$= 2^{12} + 1 = 4096 + 1 = 4097$

Q 26 :
$Let A = [\begin{matrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{matrix}], where a, c \in ℝ . If A^{3} = A and the positive value of a belongs to the interval$ $(n - 1, n],$

$where n \in ℕ, then n is equal to$ _________ . [2023]

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(2)

We have,

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

Now, $A^{2} = [\begin{matrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{matrix}] [\begin{matrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{matrix}] \begin{matrix} \end{matrix}$

$\Rightarrow A^{2} = [\begin{matrix} a + 2 & 2 c & 3 \\ 3 & a + 3 c & 2 a \\ a c & 1 & 2 + 3 c \end{matrix}] \begin{matrix} \end{matrix}$

and $A^{3} = [\begin{matrix} a + 2 & 2 c & 3 \\ 3 & a + 3 c & 2 a \\ a c & 1 & 2 + 3 c \end{matrix}] [\begin{matrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{matrix}]$

$\Rightarrow A^{3} = [\begin{matrix} 2 a c + 3 & a + 2 + 3 c & 2 a + 4 + 6 c \\ a (a + 3 c) + 2 a & 3 + 2 a c & 6 + 3 a + 9 c \\ a + 2 + 3 c & a c + c (2 + 3 c) & 2 a c + 3 \end{matrix}]$

Given that $A^{3} = A$ ,

$2 a c + 3 = 0 ...(i)$

and $a + 2 + 3 c = 1 \Rightarrow a + 1 + 3 c = 0 ...(ii)$

$\Rightarrow a + 1 - \frac{9}{2 a} = 0$ [From (i)]

$\Rightarrow 2 a^{2} + 2 a - 9 = 0$

Since $f (1) < 0, f (2) > 0$

$∴ a \in (1, 2]$ $[∵ f (a) = 2 a^{2} + 2 a - 9]$

Now, $n - 1 = 1 \Rightarrow n = 2$

Q 27 :
$Let A be a symmetric matrix such that | A | = 2 and$ $[\begin{matrix} 2 & 1 \\ 3 & \frac{3}{2} \end{matrix}]$ $A = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}] .$ If the sum of the diagonal elements of A is $s$ , then $\frac{β s}{α^{2}}$ is equal to ________ . [2023]

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(5)

Given, $| A | = 2$

Let $A = [\begin{matrix} a & b \\ b & c \end{matrix}] \Rightarrow a c - b^{2} = 2 ...(i)$

Also, $[\begin{matrix} 2 & 1 \\ 3 & \frac{3}{2} \end{matrix}] [\begin{matrix} a & b \\ b & c \end{matrix}] = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}]$

$\Rightarrow [\begin{matrix} 2 a + b & 2 b + c \\ 3 a + \frac{3}{2} b & 3 b + \frac{3}{2} c \end{matrix}] = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}]$

$\Rightarrow 2 a + b = 1 ...(ii), 2 b + c = 2 ...(iii)$

Solving (i), (ii), and (iii) we get $b = - \frac{1}{2}, a = \frac{3}{4}, c = 3$

Now, $α = 3 a + \frac{3}{2} b = \frac{3}{2}, β = 3 b + \frac{3}{2} c = 3$

$s = a + c = \frac{15}{4}$ $∴$ $\frac{β s}{α^{2}} = \frac{3 \times \frac{15}{4}}{\frac{9}{4}} = 5$

Q 28 :
Let $A = [a_{i j}], a_{i j} \in Z \cap [0, 4], 1 \leq i, j \leq 2$ . The number of matrices A such that the sum of all entries is a prime number $p \in (2, 13)$ is ________ . [2023]

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(204)

As given $a + b + c + d = 3 or 5 or 7 or 11$

If sum = 3, then

Coefficient of $x^{3}$ in ${(1 + x + x^{2} + \dots + x^{4})}^{4}$ is ${(1 - x^{5})}^{4} {(1 - x)}^{- 4}$

$∴$ ${}^{4 + 3 - 1}{C_{3} = {}^{6}{C_{3}}} = 20$

If sum = 5, coefficient of $x^{5}$ is $(1 - 4 x^{5}) {(1 - x)}^{- 4}$

$∴$ ${}^{4 + 5 - 1}{C_{5}} - 4 = {}^{8}{C_{5}} - 4 = 52$

If sum = 7, then coefficient of $x^{7}$ is

$(1 - 4 x^{5}) {(1 - x)}^{- 4} \Rightarrow {}^{10}C_{7} - 4 \times {}^{5}{C_{2}} = 80$

If sum = 11

$(1 - 4 x^{5} + 6 x^{10}) {(1 - x)}^{- 4}$ $\Rightarrow {}^{4 + 11 - 1}{C_{11}} - 4 {}^{4 + 6 - 1}{C_{6}} + 6 {}^{4 + 1 - 1}{C_{1}}$

$= {}^{14}{C_{11}} - 4$ ${}^{9}{C_{6}} + 24 = 364 - 336 + 24 = 52$

$∴ Total matrices = 20 + 52 + 80 + 52 = 204$

Topic Question Set

Q 21 :
If A and B are two non-zero $n \times n$ matrices such that $A^{2} + B = A^{2} B,$ then [2023]

Q 22 :
Let $A = [\begin{matrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{- 3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{matrix}] and B = [\begin{matrix} 1 & - i \\ 0 & 1 \end{matrix}],$ where $i = \sqrt{- 1}$ . If $M = A^{T} B A$ , then the inverse of the matrix $A M^{2023} A^{T}$ is [2023]

(3)

Q 23 :
Let A, B, C be $3 \times 3$ matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements:

$(S1) A^{13} B^{26} - B^{26} A^{13} is symmetric$

$(S2) A^{26} C^{13} - C^{13} A^{26} is symmetric$

Then, [2023]

Q 24 :
Let $α$ and $β$ be real numbers. Consider a $3 \times 3$ matrix A such that $A^{2} = 3 A + α I .$ If $A^{4} = 21 A + β I,$ then [2023]

Q 25 :
Let $A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}) .$ Then the sum of the diagonal elements of the matrix ${(A + I)}^{11}$ is equal to:

Q 26 :
$Let A = [\begin{matrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{matrix}], where a, c \in ℝ . If A^{3} = A and the positive value of a belongs to the interval$ $(n - 1, n],$

$where n \in ℕ, then n is equal to$ _________ . [2023]

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Q 27 :
$Let A be a symmetric matrix such that | A | = 2 and$ $[\begin{matrix} 2 & 1 \\ 3 & \frac{3}{2} \end{matrix}]$ $A = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}] .$ If the sum of the diagonal elements of A is $s$ , then $\frac{β s}{α^{2}}$ is equal to ________ . [2023]

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Q 28 :
Let $A = [a_{i j}], a_{i j} \in Z \cap [0, 4], 1 \leq i, j \leq 2$ . The number of matrices A such that the sum of all entries is a prime number $p \in (2, 13)$ is ________ . [2023]

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Topic Question Set

Q 21 : If A and B are two non-zero n×n matrices such that A2+B=A2B, then [2023]

Q 22 : Let A=[110310-310110] and B=[1-i01], where i=-1. If M=ATBA, then the inverse of the matrix AM2023AT is [2023]

(3)

Q 23 : Let A, B, C be 3×3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements: (S1) A13B26-B26A13 is symmetric (S2) A26C13-C13A26 is symmetric Then, [2023]

Q 24 : Let α and β be real numbers. Consider a 3×3 matrix A such that A2=3A+αI. If A4=21A+βI, then [2023]

Q 25 : Let A=(10004-1012-3). Then the sum of the diagonal elements of the matrix (A+I)11 is equal to:

Q 26 : Let A=[012a031c0], where a,c∈ℝ. If A3=A and the positive value of a belongs to the interval (n-1,n], where n∈ℕ, then n is equal to _________ . [2023]

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Q 27 : Let A be a symmetric matrix such that |A|=2 and [21332]A=[12αβ]. If the sum of the diagonal elements of A is s, then βsα2 is equal to ________ . [2023]

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Q 28 : Let A=[aij], aij∈Z∩[0,4], 1≤i,j≤2. The number of matrices A such that the sum of all entries is a prime number p∈(2,13) is ________ . [2023]

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Q 21 :
If A and B are two non-zero $n \times n$ matrices such that $A^{2} + B = A^{2} B,$ then [2023]

Q 22 :
Let $A = [\begin{matrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{- 3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{matrix}] and B = [\begin{matrix} 1 & - i \\ 0 & 1 \end{matrix}],$ where $i = \sqrt{- 1}$ . If $M = A^{T} B A$ , then the inverse of the matrix $A M^{2023} A^{T}$ is [2023]

Q 23 :
Let A, B, C be $3 \times 3$ matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements:

$(S1) A^{13} B^{26} - B^{26} A^{13} is symmetric$

$(S2) A^{26} C^{13} - C^{13} A^{26} is symmetric$

Then, [2023]

Q 24 :
Let $α$ and $β$ be real numbers. Consider a $3 \times 3$ matrix A such that $A^{2} = 3 A + α I .$ If $A^{4} = 21 A + β I,$ then [2023]

Q 25 :
Let $A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 4 & - 1 \\ 0 & 12 & - 3 \end{matrix}) .$ Then the sum of the diagonal elements of the matrix ${(A + I)}^{11}$ is equal to:

Q 26 :
$Let A = [\begin{matrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{matrix}], where a, c \in ℝ . If A^{3} = A and the positive value of a belongs to the interval$ $(n - 1, n],$

$where n \in ℕ, then n is equal to$ _________ . [2023]

Q 27 :
$Let A be a symmetric matrix such that | A | = 2 and$ $[\begin{matrix} 2 & 1 \\ 3 & \frac{3}{2} \end{matrix}]$ $A = [\begin{matrix} 1 & 2 \\ α & β \end{matrix}] .$ If the sum of the diagonal elements of A is $s$ , then $\frac{β s}{α^{2}}$ is equal to ________ . [2023]

Q 28 :
Let $A = [a_{i j}], a_{i j} \in Z \cap [0, 4], 1 \leq i, j \leq 2$ . The number of matrices A such that the sum of all entries is a prime number $p \in (2, 13)$ is ________ . [2023]