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JEE • MATHEMATICS • MATRICES

Matrices practice · Easy

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211 Easy
211 of 922
Analyzing +4 | -1
If Δ=∣cos⁡α−sin⁡α1sin⁡αcos⁡α1cos⁡(α+β)−sin⁡(α+β)1∣,\Delta=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|, then
View answer and explanation
Correct option: A
Δ∈[1−2,1+2]\Delta \in[1-\sqrt{2}, 1+\sqrt{2}]
The correct answer is A (Δ∈[1−2,1+2]\Delta \in[1-\sqrt{2}, 1+\sqrt{2}]) because the determinant simplifies to 1+sin⁡β−cos⁡β1 + \sin \beta - \cos \beta, and the range of sin⁡β−cos⁡β\sin \beta - \cos \beta is [−2,2][-\sqrt{2}, \sqrt{2}].
212 Easy
212 of 922
Application +4 | -1
If the matrix [abcd]\left[ {\begin{array}{cccccccccccccccccccc} a&b\\ c&d \end{array}} \right] is commutative w.r.t multiplication with the matrix [1101],\left[ {\begin{array}{cccccccccccccccccccc} 1&1\\ 0&1 \end{array}} \right], then
View answer and explanation
Correct option: C
c=0,d=ac = 0,d = a
Option C (c=0,d=ac = 0, d = a) is correct because it satisfies the condition for matrix commutativity with the given matrix.
213 Easy
213 of 922
Analyzing +4 | -1
The total number of matrices
A=[02y12xy−12x−y1],(x,y∈R,x≠y)A = \left[ {\begin{array}{cccccccccccccccccccc} 0&{2y}&1\\ {2x}&y&{ - 1}\\ {2x}&{ - y}&1 \end{array}} \right],(x,y \in R,x \ne y) for which ATA=3I3{A^T}A = 3{I_3} is
View answer and explanation
Correct option: B
4
Option B (4) is correct because there are 4 possible matrices that satisfy the given condition ATA=3I3{A^T}A = 3{I_3}.
214 Easy
214 of 922
Analyzing +4 | -1
The set of equations λx−y+(cos⁡θ)z=0\lambda x-y+(\cos \theta) z=0, 3x+y+2z=0,(cos⁡θ)x+y+2z=0,0≤θ≤2π3 x+y+2 z=0,(\cos \theta) x+y+2 z=0,0 \leq \theta \leq 2 \pi, has nontrivial solution(s)
View answer and explanation
Correct option: A
for no value of λ\lambda and θ\theta
Option A (for no value of λ\lambda and θ\theta) is correct because the condition for a non-trivial solution leads to cos⁡θ=3\cos \theta = 3 or cos⁡θ=−2\cos \theta = -2, which are impossible, so there are no values of λ\lambda and θ\theta that give non-trivial solutions.
215 Easy
215 of 922
Understanding +4 | -1
If A,BA, B are two n×nn \times n non-singular matrices, then
View answer and explanation
Correct option: A
ABA B is non-singular
Option A (ABA B is non-singular) is correct because if AA and BB are non-singular, then ∣A∣≠0|A| \neq 0 and ∣B∣≠0|B| \neq 0, so ∣AB∣=∣A∣∣B∣≠0|AB| = |A||B| \neq 0, which means ABAB is non-singular.
216 Easy
216 of 922
Analyzing +4 | -1
The value of the determinant Δ=∣sin⁡2αsin⁡(α+β)sin⁡(α+γ)sin⁡(β+γ)sin⁡2βsin⁡(γ+β)(sin⁡γ+α)sin⁡(γ+β)sin⁡2γ∣,\Delta=\left|\begin{array}{rrr}\sin 2 \alpha & \sin (\alpha+\beta) & \sin (\alpha+\gamma) \\ \sin (\beta+\gamma) & \sin 2 \beta & \sin (\gamma+\beta) \\ (\sin \gamma+\alpha) & \sin (\gamma+\beta) & \sin 2 \gamma\end{array}\right|, is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the determinant can be expressed as a product of two determinants, each of which is zero.
217 Easy
217 of 922
Application +4 | -1
The determinant Δ=∣11+ii1+ii1i11+i∣ equals\Delta=\left|\begin{array}{ccc} 1 & 1+i & i \\ 1+i & i & 1 \\ i & 1 & 1+i \end{array}\right| \text { equals}
View answer and explanation
Correct option: D
2(i−1)2(i-1)
Option D (2(i-1)) is correct because after simplifying the determinant using column operations, the result is 2(i-1).
218 Easy
218 of 922
Application +4 | -1
If A=[100010ab−1]A = \left[ {\begin{array}{cccccccccccccccccccc} 1&0&0\\ 0&1&0\\ a&b&{ - 1} \end{array}} \right], then A2{A^2} is equal to
View answer and explanation
Correct option: B
Unit matrix
Option B (Unit matrix) is correct because when matrix A is multiplied by itself, the result is the identity matrix.
219 Easy
219 of 922
Application +4 | -1
If ω\omega is a complex cube root of unity and A=[ω    00      ω]A = \left[ {\begin{array}{cccccccccccccccccccc} {\omega \;\;0}\\ {0\;\;\;\omega } \end{array}} \right], then A50  {A^{50}}\; is
View answer and explanation
Correct option: A
ω2A{\omega ^2}A
The correct answer is A. Since ω\omega is a complex cube root of unity, ω3=1\omega^3 = 1. Using this property, A50A^{50} can be simplified to ω2A\omega^2 A.
220 Easy
220 of 922
Analyzing +4 | -1
Let AA and BB be 3×33 \times 3 matrices of real numbers, where AA is symmetric, B is skew-symmetric, and (A+B)(A−B)=(A−B)(A+B)(A+B)(A-B)=(A-B)(A+B) if (AB)T=(−1)nAB(A B)^T=(-1)^n A B, then
View answer and explanation
Correct option: D
nn is an odd natural number
The correct answer is nn is an odd natural number because the given conditions lead to (−1)n=−1(-1)^n = -1, which is only true for odd n.
221 Easy
221 of 922
Application +4 | -1
If ∣x218x∣=∣62186∣\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\ 18 & 6\end{array}\right|, then x\mathrm{x} is cqual to
View answer and explanation
Correct option: B
±6\pm 6
Option B (±6\pm 6) is correct because solving the equation derived from the determinant equality yields two possible values for x: 6 and -6.
222 Easy
222 of 922
Application +4 | -1
The value of Δ=∣1ab+c1bc+a1ca+b∣,\Delta=\left|\begin{array}{lll}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{array}\right|, is
View answer and explanation
Correct option: D
0
Option D (0) is correct because applying the column operation C2 -> C2 + C3 results in two identical columns, making the determinant zero.
223 Easy
223 of 922
Understanding +4 | -1
If ∣1+ax1+bx1+cx1+a1x1+b1x1+c1x1+a2x1+b2x1+c2x∣=A0+A1x+A2x2+A3x3,then  A0\left| {\begin{array}{cccccccccccccccccccc} {1 + ax}&{1 + bx}&{1 + cx}\\ {1 + {a_1}x}&{1 + {b_1}x}&{1 + {c_1}x}\\ {1 + {a_2}x}&{1 + {b_2}x}&{1 + {c_2}x} \end{array}} \right| = {A_0} + {A_1}x + {A_2}{x^2} + {A_3}{x^3},\mathrm{then}\,\,{A_0} is equal to
View answer and explanation
Correct option: B
0
Option B is correct because when x=0, the determinant has two identical rows, making its value zero.
224 Easy
224 of 922
Analyzing +4 | -1
The system of equations 2x+3y+2z=12 x+3 y+2 z=1, 4x+6y+2z=1,−x+y+2z=34 x+6 y+2 z=1,-x+y+2 z=3 has
View answer and explanation
Correct option: A
a unique solution
Option A is correct because the determinant of the coefficient matrix is non-zero, indicating a unique solution.
225 Easy
225 of 922
Analyzing +4 | -1
The matrix [5103−2−46−2−2b]\left[ {\begin{array}{cccccccccccccccccccc} 5&{10}&3\\ { - 2}&{ - 4}&6\\ { - 2}&{ - 2}&b \end{array}} \right] is a singular matrix, if bb is equal to
View answer and explanation
Correct option: D
For any value of bb
The correct answer is D (For any value of b) because the determinant of the matrix is always zero regardless of the value of b, making it a singular matrix.

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
226 Easy
226 of 922
Analyzing +4 | -1
If Δ1=∣b+ca−bac+ab−cba+bc−ac∣\Delta_1=\left|\begin{array}{lll}b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c\end{array}\right| and Δ2=∣abcbcacab∣, then Δ1−Δ2 equal\Delta_2=\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| \text {, then } \Delta_1-\Delta_2 \text { equal}
View answer and explanation
Correct option: A
0
Option A (0) is correct because after applying column operations, the determinant Δ1\Delta_1 simplifies to Δ2\Delta_2, making the difference zero.
227 Easy
227 of 922
Application +4 | -1
Let AA be the set of all 3×33 \times 3 symmetric matrices all of whose entries are either 0 or 1 such that five of these entries are 1 and four are 0 . The number of matrices in set AA, is
View answer and explanation
Correct option: A
12
Option A (12) is correct because there are 3 matrices with three 1s on the diagonal and 9 matrices with one 1 on the diagonal, summing to 12.
228 Easy
228 of 922
Understanding +4 | -1
If AA is an invertible matrix, then det⁡(A−1)\det \left( {{A^{ - 1}}} \right) is equal to
View answer and explanation
Correct option: B
1det⁡(A)\frac{1}{{\det \left( A \right)}}
Option B is correct because the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.
229 Easy
229 of 922
Understanding +4 | -1
If AA is singular matrix, then A  adj  (A)A\;\mathrm{adj}\;\left( A \right)
View answer and explanation
Correct option: B
Is a zero matrix
Option B (Is a zero matrix) is correct because for a singular matrix A, its determinant is 0, and A adj(A) equals |A|I, which results in a zero matrix.
230 Easy
230 of 922
Application +4 | -1
If A=[cos⁡θsin⁡θsin⁡θ−cos⁡θ],B=[10−11],C=ABAT,A=\left[\begin{array}{rr}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta\end{array}\right], B=\left[\begin{array}{rr}1 & 0 \\ -1 & 1\end{array}\right], C=A B A^T, then ATCnA,n∈I+A^T C^n A, n \in I^{+}equals to
View answer and explanation
Correct option: D
[10−n1]\left[\begin{array}{rr}1 & 0 \\ -n & 1\end{array}\right]
Option D, [10−n1]\left[\begin{array}{rr}1 & 0 \\ -n & 1\end{array}\right], is correct because after repeated application of the given transformations, ATCnAA^T C^n A simplifies to BnB^n, which equals the given matrix.
231 Easy
231 of 922
Application +4 | -1
If CC is a 3×33 \times 3 matrix satisfying the relation C2+C=IC^2+C=I, then C−2C^{-2} is given by
View answer and explanation
Correct option: D
2I+C2 I+C
Option D (2I+C) is correct because by manipulating the given equation and using the inverse, we can express C-2 as 2I+C.
232 Easy
232 of 922
Understanding +4 | -1
If A\mathrm{A} is a symmetric matrix, then matrix MrAM\mathrm{M}^r \mathrm{AM} is
View answer and explanation
Correct option: A
Symmetnic
Option A (Symmetric) is correct because the transpose of M'AM is equal to itself, indicating symmetry.
233 Easy
233 of 922
Application +4 | -1
If AA is a skew-symmetric matrix of order 3 , then the matrix A4A^4 is
View answer and explanation
Correct option: B
Symmetric matrix
Option B (Symmetric matrix) is correct because the transpose of A4A^4 is equal to A4A^4, which is the definition of a symmetric matrix.
234 Easy
234 of 922
Application +4 | -1
If A=[22−32],B=[0−110],A = \left[ {\begin{array}{cccccccccccccccccccc} 2&2\\ { - 3}&2 \end{array}} \right],B = \left[ {\begin{array}{cccccccccccccccccccc} 0&{ - 1}\\ 1&0 \end{array}} \right], then (B−1A−1)−1={\left( {{B^{ - 1}}{A^{ - 1}}} \right)^{ - 1}} =
View answer and explanation
Correct option: A
[2−223]\left[ {\begin{array}{cccccccccccccccccccc} 2&{ - 2}\\ 2&3 \end{array}} \right]
Option A is correct. The expression simplifies to AB, and the matrix multiplication results in the given matrix.
235 Easy
235 of 922
Application +4 | -1
If x[−34]+y[43]=[10−5],  thenx\left[ {\begin{array}{cccccccccccccccccccc} { - 3}\\ 4 \end{array}} \right] + y\left[ {\begin{array}{cccccccccccccccccccc} 4\\ 3 \end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc} {10}\\ { - 5} \end{array}} \right],\;\mathrm{then}
View answer and explanation
Correct option: A
x=−2,  y=1x = - 2,\;y = 1
Option A (x=−2,  y=1x = - 2,\;y = 1) is correct because these values satisfy the given matrix equation.
236 Easy
236 of 922
Application +4 | -1
If A=[abcd]A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] such that AA satisfies the relation A2−(a+d)A=OA^2-(a+d) A=O, then inverse of AA is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because if A has an inverse and satisfies the given relation, then all elements of A must be zero, which contradicts the existence of an inverse. Therefore, A-1 does not exist.
237 Easy
237 of 922
Application +4 | -1
Let θ=π5\theta=\frac{\pi}{5} and A=[cos⁡θsin⁡θ−sin⁡θcos⁡θ]A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]. If B=A+A4B=A+\mathbf{A}^4, then det⁡(B)\operatorname{det}(\mathrm{B}) :
View answer and explanation
Correct option: D
lies in (1,2)(1,2)
Option C (log⁡e2\log _{e} 2) is correct because the limit can be expressed as a definite integral which evaluates to log⁡e2\log_e 2.
238 Easy
238 of 922
Application +4 | -1
Let A=[1a01]A=\left[\begin{array}{ll}1 & a \\ 0 & 1\end{array}\right], where a>0a > 0. Sum of the series⁡S=trace⁡(A)+trace⁡(12A)+trace⁡(122A2)+trace⁡(123A3)\operatorname{series} S=\operatorname{trace}(A)+\operatorname{trace}\left(\frac{1}{2} A\right)+\operatorname{trace}\left(\frac{1}{2^2} A^2\right)+\operatorname{trace}\left(\frac{1}{2^3} A^3\right) +…+\ldots is
View answer and explanation
Correct option: B
4
Option B (4) is correct because the sum of the series can be found by recognizing the pattern in the trace of the matrix and summing the resulting geometric series.
239 Easy
239 of 922
Analyzing +4 | -1
If AA and BB are different matrices satisfying A3=B3A^3=B^3 and A2B=B2A,A^2 B=B^2 A, then
View answer and explanation
Correct option: D
atleast one of det⁡(A2+B2)\operatorname{det}\left(A^2+B^2\right) or det⁡(A−B)\operatorname{det}(A-B) must be zero
Option D (at least one of det(A2+B2) or det(A-B) must be zero) is correct because the given conditions lead to (A2+B2)(A-B) = 0, implying at least one determinant must be zero.
240 Easy
240 of 922
Analyzing +4 | -1
The value of the determinant ∣b+ca−bac+ab−cba+bc−ac∣,\left|\begin{array}{lll}b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c\end{array}\right|, is
View answer and explanation
Correct option: B
3abc−a3−b3−c33 a b c-a^3-b^3-c^3
Option B (3abc−a3−b3−c33abc - a^3 - b^3 - c^3) is correct because the determinant simplifies to this expression after applying column and row operations.
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