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JEE • MATHEMATICS • MATRICES

Matrices practice · Mixed difficulty

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631 Easy
631 of 995
Analyzing +4 | -1
The rank of matrix [4      1      0      03      0      1      06      0    2      0]\left[ {\begin{array}{cccccccccccccccccccc} {4\;\;\;1\;\;\;0\;\;\;0}\\ {3\;\;\;0\;\;\;1\;\;\;0}\\ {6\;\;\;0\;\;2\;\;\;0} \end{array}} \right] is
View answer and explanation
Correct option: C
2
Option C (2) is correct because the rank is determined by finding the largest non-zero minor of the matrix.
632 Easy
632 of 995
Analyzing +4 | -1
If a,b,ca,b,c be respectively the pth,qthp\mathrm{th},q\mathrm{th} and rthr\mathrm{th} terms of a HP., then

Δ  =∣bccaabpqr111∣\Delta \; = \left| {\begin{array}{cccccccccccccccccccc} {bc}&{ca}&{ab}\\ p&q&r\\ 1&1&1 \end{array}} \right| equals
View answer and explanation
Correct option: B
00
Option B (00) is correct because the determinant is zero, which is shown by expressing the reciprocals of the HP terms as an AP and manipulating the determinant.
633 Medium
633 of 995
Application +4 | -1
If MM is a 3×33 \times 3 matrix such that M2=OM^2=O, then det⁡.((I+M)50−50M)\operatorname{det} .\left((I+M)^{50}-50 M\right) where I{I} is an identity matrix of order 3, is equal to:
View answer and explanation
Correct option: D
1
Option D (1) is correct because, given M²=O, the expression simplifies to det(I), which equals 1.
634 Hard
634 of 995
Application +4 | -1
Let α   and     β\alpha \,\,\,\mathrm{and}\,\,\,\,\,\beta be the roots of equation x2−6x−2=0.{x^2} - 6x - 2 = 0. If an=αn−βn,{a_n} = {\alpha ^n} - {\beta ^n}, for n≥1n \ge 1 then the value of a10−2a82a9\frac{{{a_{10}} - 2{a_8}}}{{2{a_9}}} is equal to
View answer and explanation
Correct option: C
3
Option C (3) is correct because, after substituting the roots into the given expression and simplifying using the quadratic equation, the value reduces to 3.
635 Easy
635 of 995
Application +4 | -1
AA is an involutory matrix given by A=[01−14−343−34],A=\left[\begin{array}{rrr}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4\end{array}\right], the inverse of A2\frac{A}{2} will be
View answer and explanation
Correct option: A
2 A2 \mathrm{~A}
Option A (2A) is correct because for an involutory matrix A, A = A-1, so (A/2)-1 = 2A-1 = 2A.
636 Easy
636 of 995
Understanding +4 | -1
For any square matrix A,  AATA,\;A{A^T} is a
View answer and explanation
Correct option: B
Symmetric matrix
Option B (Symmetric matrix) is correct because the transpose of AA^T is equal to itself, satisfying the definition of a symmetric matrix.
637 Easy
637 of 995
Application +4 | -1
If Δ=∣−a2b00−a2b2b0−a∣=0\Delta=\left|\begin{array}{ccc}-a & 2 b & 0 \\ 0 & -a & 2 b \\ 2 b & 0 & -a\end{array}\right|=0, then
View answer and explanation
Correct option: D
a/ba/b is a cube root of 8
Option D (a/ba/b is a cube root of 8) is correct because the determinant equals zero when (a/b)3 = 8, meaning a/b is a cube root of 8.
638 Easy
638 of 995
Application +4 | -1
If the matrix Mr{M_r} is given by Mr=[rr−1r−1r],=1,2,3,.…,{M_r} = \left[ {\begin{array}{cccccccccccccccccccc} r&{r - 1}\\ {r - 1}&r \end{array}} \right], = 1,2,3{,_.} \ldots , then the value of det (M1)+det⁡(M2)+.…+det⁡(M2008)\left( {{M_1}} \right) + \det \left( {{M_2}} \right){ + _.} \ldots + \det \left( {{M_{2008}}} \right) is
View answer and explanation
Correct option: C
(2008)2{\left( {2008} \right)^2}
Option C ((2008)2(2008)^2) is correct because the sum of the determinants forms an arithmetic progression, and its sum equals (2008)2(2008)^2.
639 Easy
639 of 995
Understanding +4 | -1
If a matrix AA is both symmetric and skew-symmetric, then
View answer and explanation
Correct option: C
AA is a zero matrix
Option C (AA is a zero matrix) is correct because if a matrix is both symmetric and skew-symmetric, it must be the zero matrix.
640 Easy
640 of 995
Application +4 | -1
The rank of ∣1−1111−1−111∣\left| {\begin{array}{cccccccccccccccccccc} 1&{ - 1}&1\\ 1&1&{ - 1}\\ { - 1}&1&1 \end{array}} \right| is
View answer and explanation
Correct option: D
3
Option D is correct. The determinant of the matrix is non-zero, indicating that the rank of the matrix is 3.
641 Easy
641 of 995
Application +4 | -1
The values of λ\lambda for which the matrix A=(λ0λλ0−λ010)A=\left(\begin{array}{ccc}\lambda & 0 & \lambda \\ \lambda & 0 & -\lambda \\ 0 & 1 & 0\end{array}\right) is orthogonal is
View answer and explanation
Correct option: D
±1/2\pm 1 / \sqrt{2}
Option D (±1/2\pm 1 / \sqrt{2}) is correct because for the matrix to be orthogonal, A*A' = I, which leads to the condition λ=±1/2\lambda = \pm 1 / \sqrt{2}.
642 Medium
642 of 995
Application +4 | -1
If α,β,γ\alpha, \beta, \gamma are the roots of x3+ax2+b=0x^3+a x^2+b=0, then the determinant Δ=∣αβγβγαγαβ∣ equals\Delta=\left|\begin{array}{lll} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right| \text { equals}
View answer and explanation
Correct option: D
a3a^3
The correct answer is D (a3a^3) because by applying column and row operations and using Vieta's formulas, the determinant simplifies to a3a^3.
643 Easy
643 of 995
Application +4 | -1
The matrix A=[12              22−12    −12]A = \left[ {\begin{array}{cccccccccccccccccccc} {\frac{1}{{\sqrt {2\;\;} }}\;\;\;\;\;\frac{2}{{\sqrt 2 }}}\\ { - \frac{1}{{\sqrt 2 }}\;\; - \frac{1}{{\sqrt 2 }}} \end{array}} \right] is
View answer and explanation
Correct option: C
Nilpotent
Option C (Nilpotent) is correct because A2 results in a zero matrix, satisfying the definition of a nilpotent matrix.
644 Easy
644 of 995
Analyzing +4 | -1
If a>b>ca>b>c and the system of equations ax+by+cz=0,bx+cy+az=0a x+b y+c z=0, b x+c y+a z=0 and cx+ay+bz=0c x+a y+b z=0 has a non-trivial solution, then the quadratic equation ax2+bx+c=0ax^2+b x+c=0 has
View answer and explanation
Correct option: A
At least one positive root
Option A (At least one positive root) is correct because the condition a+b+c=0 implies that x=1 is a root of the quadratic equation ax²+bx+c=0, which is a positive root.
645 Medium
645 of 995
Analyzing +4 | -1
For the system of linear equations

x+y+z=6αx+βy+7z=3x+2y+3z=14,\begin{array}{l} x+y+z=6 \\ \alpha x+\beta y+7 z=3 \\ x+2 y+3 z=14,\end{array}
which of the following is NOT true?
View answer and explanation
Correct option: D
For every point (α,β)≠(7,7)(\alpha, \beta) \neq(7,7) on the line x−2y+7=0x-2 y+7=0, the system has infinitely many solutions
Option D is correct because the condition for infinitely many solutions is not satisfied for every point on the line x−2y+7=0x - 2y + 7 = 0 except (7,7).

Explore Other Chapters in MATHEMATICS

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs
646 Easy
646 of 995
Application +4 | -1
If XX is a 2×32 \times 3 matrix such that [X′X]≠0\left[X^{\prime} X\right] \neq 0, and AA =I−X(X′X)−1X′=I-X\left(X^{\prime} X\right)^{-1} X^{\prime} then A2A^2 is equal to
View answer and explanation
Correct option: A
AA
Option A (A) is correct because by expanding A2 and using the given expression for A, we find that A2 = A.
647 Medium
647 of 995
Application +4 | -1
If matrix P=[0−tan⁡(θ/2)tan⁡θ/20]P=\left[\begin{array}{cc}0 & -\tan (\theta / 2) \\ \tan \theta / 2 & 0\end{array}\right], then find
(I−P)[cos⁡θ−sin⁡θsin⁡θcos⁡θ]({I}-{P})\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]
View answer and explanation
Correct option: C
I+P{I}+{P}
Option C (I+P{I}+{P}) is correct because after performing the matrix multiplication and applying trigonometric identities, the result simplifies to I+P{I}+{P}.
648 Easy
648 of 995
Understanding +4 | -1
If for the matrix A,A3=I\mathrm{A}, \mathrm{A}^3=\mathrm{I}, then A−1=\mathrm{A}^{-1}=
View answer and explanation
Correct option: A
A2\mathrm{A}^2
Option A (A2) is correct because if A3 = I, multiplying both sides by A-1 gives A2 = A-1.
649 Medium
649 of 995
Application +4 | -1
Let AA be a matrix such that A.[1203]A .\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right] is a scalar matrix and ∣3A∣=108|3 A|=108. Then A2A^2 equals]
View answer and explanation
Correct option: D
[36−3204]\left[\begin{array}{cc}36 & -32 \\ 0 & 4\end{array}\right]
Option D (\left[\begin{{cc}}36 & -32 \\ 0 & 4\end{{cc}}\right]\left[\begin{{cc}}36 & -32 \\ 0 & 4\end{{cc}}\right]) is correct. By finding the matrix A such that A multiplied by the given matrix results in a scalar matrix, and using the determinant condition, we can find A².
650 Medium
650 of 995
Analyzing +4 | -1
If Δ=∣13cos⁡θ1sin⁡θ13cos⁡θ1sin⁡θ1∣\Delta=\left|\begin{array}{ccc}1 & 3 \cos \theta & 1 \\ \sin \theta & 1 & 3 \cos \theta \\ 1 & \sin \theta & 1\end{array}\right|, then the maximum value of Δ\Delta is:
View answer and explanation
Correct option: C
10
Option C (10) is correct because the maximum value of the determinant is found by simplifying the expression and maximizing the resulting trigonometric function.
651 Easy
651 of 995
Understanding +4 | -1
If AA and BB are two matrices of the order 3×m3 \times m and 3×n3 \times n, respectivelyand m=nm=n, then order of mamtrix (5A−2B)(5 A-2 B) is
View answer and explanation
Correct option: D
3×n3 \times n
Option D (3 x n) is correct because if m=n, both matrices A and B have the same order (3 x n), and the resulting matrix (5A - 2B) will also have the same order.
652 Medium
652 of 995
Application +4 | -1
The minors of −4- 4 and 9 and the cofactors of −4- 4 and 9 in matrix [−1−23−4−5−6−789]\left[ {\begin{array}{cccccccccccccccccccc} { - 1}&{ - 2}&3\\ { - 4}&{ - 5}&{ - 6}\\ { - 7}&8&9 \end{array}} \right] are respectively
View answer and explanation
Correct option: B
−42,−3,42,−3- 42, - 3,42, - 3
Option B (-42, -3, 42, -3) is correct because it accurately calculates the minors and cofactors of the specified elements in the matrix.
653 Easy
653 of 995
Understanding +4 | -1
If AA is 2×22 \times 2 matrix such that A2=OA^2=O, then tr⁡(A)\operatorname{tr}(A) is
View answer and explanation
Correct option: C
0
Option C (0) is correct because if A2=OA^2 = O, then the eigenvalues of AA are 0, and the trace of AA is the sum of its eigenvalues.
654 Medium
654 of 995
Application +4 | -1
If A=[cos⁡θ−sin⁡θsin⁡θcos⁡θ]A = \left[ {\begin{array}{cccccccccccccccccccc} {\cos \theta }&{ - \sin \theta }\\ {\sin \theta }&{\cos \theta } \end{array}} \right] then the matrix A−50  when  θ=π12,{A^{ - 50}}\mathrm{ }\,\,\mathrm{when }\,\,\theta = \frac{\pi }{{12}}, is equal to
View answer and explanation
Correct option: C
[3212−1232]\left[ {\begin{array}{cccccccccccccccccccc} {\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\ { - \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}} \end{array}} \right]
Option C (\left[ {\begin{{array}}{{{*{20}{c}}}}{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\{- \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\end{{array}}} \right]\left[ {\begin{{array}}{{{*{20}{c}}}}{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\{- \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}\end{{array}}} \right]) is correct because it is the result of calculating A−50{A^{ - 50}} using the given matrix A and θ=π12\theta = \frac{\pi }{{12}}.
655 Easy
655 of 995
Understanding +4 | -1
The trace Tr(A)T_r(A) of a 3×33 \times 3 matrix A=(aij)A=\left(a_{i j}\right) is defined by the relation Tr(A)=a11+a22+a33T_r(A)=a_{11}+a_{22}+a_{33} (i.e. Tr(A)T_r(A) is sum of the main diagonal elements). Which of the following statements cannot hold ?
View answer and explanation
Correct option: D
Tr(A2)=Tr(A)2\mathrm{T}_{\mathrm{r}}\left(\mathrm{A}^2\right)=\mathrm{T}_{\mathrm{r}}(\mathrm{A})^2
Option D (Tr(A2)=Tr(A)2\mathrm{T}_{\mathrm{r}}(\mathrm{A}^2)=\mathrm{T}_{\mathrm{r}}(\mathrm{A})^2) cannot hold because the trace of A squared is not generally equal to the square of the trace of A.
656 Easy
656 of 995
Analyzing +4 | -1
If Δ=∣0b−ac−aa−b0c−ba−cb−c0∣,\Delta=\left|\begin{array}{ccc}0 & b-a & c-a \\ a-b & 0 & c-b \\ a-c & b-c & 0\end{array}\right|, then Δ\Delta equals
View answer and explanation
Correct option: D
0
Option D (0) is correct because the given determinant represents a skew-symmetric matrix of odd order, and the determinant of any skew-symmetric matrix of odd order is always zero.
657 Easy
657 of 995
Understanding +4 | -1
If B\mathrm{B} is a non-singular matrix and A\mathrm{A} is a square matrix, then det⁡(B−1AB)\operatorname{det}\left(\mathrm{B}^{-1} \mathrm{AB}\right) is equal to
View answer and explanation
Correct option: C
det⁡(A)\operatorname{det}(\mathrm{A})
Option C (det(A)) is correct because the determinant of B⁻¹AB simplifies to the determinant of A using properties of determinants.
658 Medium
658 of 995
Application +4 | -1
If A{A} is an idempotent matrix satisfying (I−0.4 A)−1=I−αA(I-0.4 {~A})^{-1}={I}-\alpha {A}, where I{I} is unit matrix of the same order as that of AA, then the value of α\alpha is:
View answer and explanation
Correct option: C
−23\frac{-2}{3}
Option C (−23\frac{{-2}}{{3}}) is correct because by substituting the idempotent matrix property into the given equation and simplifying, we can solve for α\alpha.
659 Easy
659 of 995
Analyzing +4 | -1
Let M=[auv]n×nM = {\left[ {{a_{uv}}} \right]_{n \times n}} be a matrix, where auv=sin⁡(θu−θv)+icos⁡(θu−θv){a_{uv}} = \sin \left( {{\mathrm{\theta }_u} - {\mathrm{\theta }_v}} \right) + i\cos \left( {{\mathrm{\theta }_u} - {\mathrm{\theta }_v}} \right), the MM is equal to
View answer and explanation
Correct option: D
−MˉT- {\bar M^T}
Option D (−MˉT-{\bar M^T}) is correct because taking the conjugate transpose of the conjugate of M results in the negative of the original matrix M.
660 Easy
660 of 995
Application +4 | -1
If A,BA, B and CC are three square matrices of the same size such that B=CAC−1B=C A C^{-1}, then CA3C−1C A^3 C^{-1} is equal to
View answer and explanation
Correct option: C
B3B^3
Option C (B3B^3) is correct because by repeatedly substituting B = CAC-1, we can show that CA3C-1 = B3.
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