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Matrices and Determinants

56 questions · Mathematics · JEE Advanced

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PYQ Library
JEE Advanced
Mathematics
Matrices and Determinants

Matrices and Determinants

56 questions · Mathematics · JEE Advanced

Consider the matrix $P = 200020003 .$ Let the transpose of a matrix $X$ be denoted by $X^{T}$ . Then the number of $3 \times 3$ invertible matrices $Q$ with…2025 · Shift 1 · Q20 · MCQ
Let $I = (1001)$ and $P = (2003)$ . Let $Q = (x z y 4)$ for some non-zero real…2025 · Shift 2 · Q21 · Multiple correct
Let $R^{2}$ denote $R \times R$ . Let $S = {(a, b, c) : a, b, c \in R and a x^{2} + 2 b x y + c y^{2} > 0 for all (x, y) \in R^{2} - {(0, 0)}} .$ Then which of the following…2024 · Shift 1 · Q23 · Multiple correct
Let $S = ⎩ ⎨ ⎧ A = 011 1 a b c d e : a, b, c, d, e \in {0, 1}$ and $∣ A ∣ \in {- 1, 1}}$ , where $∣ A ∣$ denotes the determinant of $A$ . Then…2024 · Shift 1 · Q27 · Numerical
Let $α$ and $β$ be the distinct roots of the equation $x^{2} + x - 1 = 0$ . Consider the set $T = {1, α, β}$ . For a $3 \times 3$ matrix $M = (a_{ij})_{3 \times 3}$ , define $R_{i} = a_{i 1} + a_{i 2} + a_{i 3}$ and $C_{j} = a_{1 j} + a_{2 j} + a_{3 j}$ … Includes table2024 · Shift 1 · Q31 · MCQ
Let $γ$ and $γ$ be real numbers. Consider the following system of linear equations $x + 2 y + z = 7 x + α z = 11 2 x - 3 y + β z = γ$ Match each entry in List-I to… Includes table2023 · Shift 1 · Q31 · MCQ
Let $M = (a_{ij}), i, j \in {1, 2, 3}$ , be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j + 1$ is divisible by $i$ , otherwise $a_{ij} = 0$ . Then which of the following statements is(are) true?2023 · Shift 2 · Q22 · Multiple correct
Let $R = ⎩ ⎨ ⎧ a c 0 325 b d 0 : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19} ⎭ ⎬ ⎫$ . Then the number of invertible matrices in $R$ is :2023 · Shift 2 · Q29 · Numerical
Let $p, q, r$ be nonzero real numbers that are, respectively, the $1 0^{th}, 10 0^{th}$ and $100 0^{th}$ terms of a harmonic progression. Consider the system of linear equations $x + y + z = 1 10 x + 100 y + 1000 z = 0 q r x + p r y + pq z = 0$ … Includes table2022 · Shift 1 · Q35 · MCQ
Let $β$ be a real number. Consider the matrix $A = β 23 011 1 - 2 - 2$ If $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix, then the…2022 · Shift 2 · Q24 · Numerical
If $M = (\frac{5}{2} - \frac{3}{2} \frac{3}{2} - \frac{1}{2})$ , then which of the following matrices is equal to $M^{2022} ?$ 2022 · Shift 2 · Q34 · MCQ
Let $α$ , $β$ and $γ$ be real numbers such that the system of linear equations x + 2y + 3z = $α$ 4x + 5y + 6z = $β$ 7x + 8y + 9z = $γ -$ 1 is consistent. Let | M | represent the determinant of the matrix $M = α β - 1 210 γ 01$ …2021 · Shift 1 · Q26 · Numerical
Let $α$ , $β$ and $γ$ be real numbers such that the system of linear equations x + 2y + 3z = $α$ 4x + 5y + 6z = $β$ 7x + 8y + 9z = $γ -$ 1 is consistent. Let | M | represent the determinant of the matrix $M = α β - 1 210 γ 01$ …2021 · Shift 1 · Q27 · Numerical
For any 3 $\times$ 3 matrix M, let | M | denote the determinant of M. Let $E = 12823133418$ , $P = 100001010$ …2021 · Shift 1 · Q30 · Multiple correct
For any 3 $\times$ 3 matrix M, let |M| denote the determinant of M. Let I be the 3 $\times$ 3 identity matrix. Let E and F be two 3 $\times$ 3 matrices such that (I $-$ EF) is invertible. If G = (I $-$ EF) $-$ 1, then which of the…2021 · Shift 1 · Q33 · Multiple correct
Let M be a 3 $\times$ 3 invertible matrix with real entries and let I denote the 3 $\times$ 3 identity matrix. If M $-$ 1 = adj(adj M), then which of the following statements is/are ALWAYS TRUE?2020 · Shift 1 · Q26 · Multiple correct
The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 $\times$ 2 matrix such that the trace of A is 3 and the trace of A3 is $-$ 18, then the value of the determinant of A is .............2020 · Shift 2 · Q22 · Numerical
Let $M = [sin^{4} θ 1 + cos^{2} θ - 1 - sin^{2} θ cos^{4} θ] = α I + β M^{- 1}$ , where $α$ = $α$ …2019 · Shift 1 · Q20 · MCQ
Let $M = 013 12 b a 31$ and adj $M = - 1 8 - 5 1 - 6 3 - 1 2 - 1$ …2019 · Shift 1 · Q29 · Multiple correct
Let x $\in$ R and let $P = 100120123$ , $Q = 20 x x 4 x x 06$ …2019 · Shift 2 · Q25 · Multiple correct
$P_{1} = I = 100010001, P_{2} = 100001010, P_{3} = 010100001, P_{4} = 001100010, P_{5} = 010001100, P_{6} = 001010100$ …2019 · Shift 2 · Q26 · Multiple correct
Suppose det $k = 0 \sum n k k = 0 \sum n^{n} C_{k} . k k = 0 \sum n^{n} C_{k} k^{2} k = 0 \sum n^{n} C_{k} 3^{k} = 0$ holds…2019 · Shift 2 · Q29 · Numerical
Let S be the set of all column matrices $b_{1} b_{2} b_{3}$ such that $b_{1}, b_{2}, b_{3} \in R$ and the system of equations (in real variables) $- x + 2 y + 5 z = b_{1} 2 x - 4 y + 3 z = b_{2} x - 2 y + 2 z = b_{3}$ …2018 · Shift 2 · Q21 · Multiple correct
Let P be a matrix of order 3 $\times$ 3 such that all the entries in P are from the set { $-$ 1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .2018 · Shift 2 · Q26 · Numerical
Which of the following is(are) NOT the square of a 3 $\times$ 3 matrix with real entries?2017 · Shift 1 · Q24 · Multiple correct
For a real number $α$ , if the system $1 α α^{2} α 1 α α^{2} α 1 x y z = 1 - 1 1$ …2017 · Shift 1 · Q26 · Numerical
How many 3 $\times$ 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of MTM is 5?2017 · Shift 2 · Q21 · MCQ
Let $P = 323 - 1 0 - 5 - 2 α 0$ , where $α \in$ R. Suppose $Q = [q_{ij}]$ is a matrix such that PQ = kl, where k $\in$ R, k $e$ …2016 · Shift 1 · Q33 · Multiple correct
The total number of distinct x $\in$ R for which $x 2 x 3 x x^{2} 4 x^{2} 9 x^{2} 1 + x^{3} 1 + 8 x^{3} 1 + 27 x^{3} = 10$ is …2016 · Shift 1 · Q34 · Numerical
Let $z = \frac{- 1 + 3 i}{2}$ , where $i = - 1$ , and r, s $\in$ {1, 2, 3}. Let $P = [(- z)^{r} z^{2 s} z^{2 s} z^{r}]$ and I be the identity…2016 · Shift 1 · Q35 · Numerical
Let $P = 1416014001$ and I be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ and $\frac{q _{31} + q _{32}}{q _{21}}$ …2016 · Shift 2 · Q32 · MCQ
Let a, $λ$ , m $\in$ R. Consider the system of linear equations ax + 2y = $λ$ 3x $-$ 2y = $μ$ Which of the following statements is(are) correct?2016 · Shift 2 · Q35 · Multiple correct
Let X and Y be two arbitrary, 3 $\times$ 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 $\times$ 3, non-zero, symmetric matrix. Then which of the following matrices is(are) skew symmetric?2015 · Shift 1 · Q37 · Multiple correct
Which of the following values of $α$ satisfy the equation $(1 - α)^{2} (2 + α)^{2} (3 + α)^{2} (1 + 2 α)^{2} (2 + 2 α)^{2} (3 + 2 α)^{2} (1 + 3 α)^{2} (2 + 3 α)^{2} (3 + 3 α)^{2} = - 648 α$ …2015 · Shift 1 · Q38 · Multiple correct
Let M be a 2 $\times$ 2 symmetric matrix with integer entries. Then, M is invertible, if2014 · Shift 1 · Q34 · Multiple correct
Let M and N be two 3 $\times$ 3 matrices such that MN = NM. Further, if M $e$ N2 and M2 = N4, then2014 · Shift 1 · Q35 · Multiple correct
For 3 × 3 matrices M and N, which of the following statement(s) is(are) NOT correct?2013 · Shift 1 · Q33 · Multiple correct
Let $ω$ be a complex cube root of unity with $ω e$ 1 and P = [pij] be a n $\times$ n matrix with pij = $ω$ i + j. Then P2 $e$ 0, when n = ?2013 · Shift 2 · Q39 · Multiple correct
Let $P = [a_{ij}]$ be a 3 $\times$ 3 matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of P is 2, then the determinant of the matrix Q is2012 · Shift 1 · Q37 · MCQ
If P is a 3 $\times$ 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 $\times$ 3 identity matrix, then there exists a column matrix $X = x y z e 000$ …2012 · Shift 2 · Q36 · MCQ
If the ad joint of a 3 $\times$ 3 matrix P is $121411473$ , then the possible value(s) of the determinant of P is(are)2012 · Shift 2 · Q39 · Multiple correct
Let M and N be two 3 $\times$ 3 non-singular skew symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M2N2(MTN) $-$ 1(MN $-$ 1)T is equal to2011 · Shift 1 · Q42 · Multiple correct
Let a, b and c be three real numbers satisfying $[a b c] 187923777 = [000]$ …2011 · Shift 1 · Q43 · MCQ
Let a, b and c be three real numbers satisfying $[a b c] 187923777 = [000]$ …2011 · Shift 1 · Q44 · MCQ
Let a, b and c be three real numbers satisfying $[a b c] 187923777 = [000]$ …2011 · Shift 1 · Q45 · MCQ
Let $ω e$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $1 ω ω^{2} a 1 ω b c 1$ ,…2011 · Shift 2 · Q35 · MCQ
Let M be a 3 $\times$ 3 matrix satisfying $M 010 = - 1 23$ , $M 1 - 1 0 = 11 - 1$ …2011 · Shift 2 · Q39 · Numerical
The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $A x y z = 100$ has exactly two distinct…2010 · Shift 1 · Q51 · MCQ
Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices : $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ …2010 · Shift 1 · Q54 · MCQ
Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices : $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ …2010 · Shift 1 · Q55 · MCQ
Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices : $T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ …2010 · Shift 1 · Q56 · MCQ
Let $k$ be a positive real number and let $A B = 2 k - 1 2 k - 2 k 2 k 1 2 k 2 k - 2 k - 1 and = 0 1 - 2 k - k 2 k - 1 0 - 2 k k 2 k 0 .$ …2010 · Shift 2 · Q32 · Numerical
Let A be the set of all 3 $\times$ 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.The number of matrices in A is2009 · Shift 1 · Q38 · MCQ
Let A be the set of all 3 $\times$ 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.The number of matrices A in A for which the system of linear equations $A x y z = 100$ …2009 · Shift 1 · Q39 · MCQ
Let A be the set of all 3 $\times$ 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.The number of matrices A in A for which the system of linear equations $A x y z = 100$ …2009 · Shift 1 · Q40 · MCQ
Consider the system of equations: $x - 2 y + 3 z = - 1 - x + y - 2 z = k x - 3 y + 4 z = 1$ Statement - 1 : The system of equations has no solution for $k e 3$ . and Statement - 2 : The determinant $1 - 1 1 3 - 2 4 - 1 k 1 e 0$ …2008 · Shift 1 · Q46 · MCQ