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JEE • MATHEMATICS • MATRICES

Matrices practice · Easy

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481 Easy
481 of 922
Analyzing +4 | -1
The number of values of kk for which the linear equations 4x+ky+2z=0,   kx+4y+z=0,  2x+2y+z=04x + ky + 2z = 0,\,\,\,kx + 4y + z = 0,\,\,2x + 2y + z = 0 Possess a non - zero solutions is
View answer and explanation
Correct option: C
2
Option C (2) is correct because the determinant of the coefficient matrix must be zero for non-zero solutions, leading to a quadratic equation with two solutions for kk.
482 Easy
482 of 922
Analyzing +4 | -1
Let A=[1432].A=\left[\begin{array}{ll}1 & 4 \\ 3 & 2\end{array}\right]. If θ\theta is the angle between the two non-zero column vectors XX such that AX=λXA X=\lambda X for some scalar λ,\lambda, then tan⁡θ\tan \theta is equal to
View answer and explanation
Correct option: C
7
Option C (7) is correct because after finding the eigenvalues and corresponding eigenvectors, the cosine of the angle between the eigenvectors is used to find the tangent.
483 Easy
483 of 922
Analyzing +4 | -1
The value of the following determinant is Δ=∣111abca3b3c3∣\Delta = \left| {\begin{array}{cccccccccccccccccccc} 1&1&1\\ a&b&c\\ {{a^3}}&{{b^3}}&{{c^3}} \end{array}} \right|
View answer and explanation
Correct option: A
(a−b)(b−c)(c−a)(a+b+c)\left( {a - b} \right)\left( {b - c} \right)\left( {c - a} \right)\left( {a + b + c} \right)
Option A ((a−b)(b−c)(c−a)(a+b+c)\left( {a - b} \right)\left( {b - c} \right)\left( {c - a} \right)\left( {a + b + c} \right)) is correct because the determinant can be factored into this form using column operations and algebraic manipulation.
484 Easy
484 of 922
Understanding +4 | -1
AA is a square matrix of order 4 and II is a unit matrix, then it is true that
View answer and explanation
Correct option: B
det (2A)=16\left( {2A} \right) = 16 det (A)\left( A \right)
Option B is correct because when a constant multiplies a matrix inside a determinant, the constant is raised to the power of the matrix's order.
485 Easy
485 of 922
Analyzing +4 | -1
Let MM and NN be two 3×33 \times 3 non-singular skew-symmetric matrices such that MN=NMM N=N M. If PTP^T denotes the transpose of PP, then M2N2(MTN)−1(MN−1)TM^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T is equal to
View answer and explanation
Correct option: C
−M2-M^2
Option C (-M2) is correct. After simplifying the expression using properties of skew-symmetric matrices and matrix transpose, the result is -M2.
486 Easy
486 of 922
Remembering +4 | -1
If AA is an orthogonal matrix, then A−1A^{-1}, equals
View answer and explanation
Correct option: B
A′{A}^{\prime}
Option B ({A}^{\'}{A}^{\'}) is correct because the inverse of an orthogonal matrix is equal to its transpose.
487 Easy
487 of 922
Application +4 | -1
If  A=[123],  B=[234]  and  C=[1502],  then  which  of   theI{\text{f}}\,\,A = \left[ {123} \right],\,\,B = \left[ \begin{gathered} 2 \quad \\ 3 \quad \\ 4 \quad \\ \end{gathered} \right]\,\,{\text{and}}\,\,C = \left[ {\begin{array}{cccccccccccccccccccc} 1&5 \\ 0&2 \end{array}} \right],\,\,{\text{then}}\,\,{\text{which}}\,\,{\text{of}}\;\,{\text{the}} following is defined
View answer and explanation
Correct option: A
AB\mathrm{AB}
Option A (AB) is correct because the number of columns in A matches the number of rows in B, making matrix multiplication AB defined.
488 Easy
488 of 922
Analyzing +4 | -1
If AA is any square matrix, then det⁡(A−AT)T\det {\left( {A - {A^T}} \right)^T} is equal to
View answer and explanation
Correct option: C
Can be 0 or a perfect square
Option C (Can be 0 or a perfect square) is correct because the determinant of a skew-symmetric matrix can be 0 (if the order is odd) or a perfect square (if the order is even).
489 Easy
489 of 922
Analyzing +4 | -1
If x2+y2+z2≠0,  x=cy+bz,  y=az+cx{x^2} + {y^2} + {z^2} \ne 0,\;x = cy + bz,\;y = az + cx and z=bx+ay,z = bx + ay, then a2+b2+c2+2abc={a^2} + {b^2} + {c^2} + 2abc =
View answer and explanation
Correct option: C
1
Option C (1) is correct because the determinant of the coefficient matrix must be zero for non-trivial solutions, leading to the given expression equaling 1.
490 Easy
490 of 922
Analyzing +4 | -1
If =[ete−tcos⁡te−tsin⁡tet−e−tcos⁡t−e−t sin⁡t−e−tsin⁡t+e−tcos⁡tet2e−tsin⁡t−2e−tcos⁡t]= \left. {\left[ {\begin{array}{cccccccccccccccccccc} {{e^t}}&{{e^{ - t}}\cos t}&{{e^{ - t}}\sin t}\\ {{e^t}}&{ - {e^{ - t}}\cos t - {e^{ - t}}\,\sin t}&{ - {e^{ - t}}\sin t + {e^{ - t}}\cos t}\\ {{e^t}}&{2{e^{ - t}}\sin t}&{ - 2{e^{ - t}}\cos t} \end{array}} \right.} \right] then AA is
View answer and explanation
Correct option: B
Invertible for every t∈Rt \in R
Option B (Invertible for every t∈Rt \in R) is correct because the determinant of A is non-zero for all t in R, implying that A is always invertible.
491 Easy
491 of 922
Application +4 | -1
If A=13[12221−2a2b]A=\frac{1}{3}\left[\begin{array}{rrr}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right] is an orthogonal matrix,
View answer and explanation
Correct option: B
a=−2,b=−1a=-2, b=-1
Option D (a=−2,b=1a=-2, b=1) is correct because for A to be orthogonal, AAT=IAA^T = I. Solving the resulting equations gives a=−2a = -2 and b=−1b = -1.
492 Easy
492 of 922
Understanding +4 | -1
The rank of the matrix A=[12344321]A=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1\end{array}\right], is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the matrix has two linearly independent rows (or columns), making its rank 2.
493 Easy
493 of 922
Application +4 | -1
If f(θ)=[cos⁡2θcos⁡θsin⁡θ−sin⁡θcos⁡θsin⁡θsin⁡2θcos⁡θsin⁡θ−cos⁡θ0],f(\theta)=\left[\begin{array}{ccc}\cos ^2 \theta & \cos \theta \sin \theta & -\sin \theta \\ \cos \theta \sin \theta & \sin ^2 \theta & \cos \theta \\ \sin \theta & -\cos \theta & 0\end{array}\right], then f(π7)f\left(\frac{\pi}{7}\right) is
View answer and explanation
Correct option: D
Non-singular
Option D (Non-singular) is correct because the determinant of the matrix f(θ) is 1, which is non-zero, indicating that the matrix is non-singular.
494 Easy
494 of 922
Application +4 | -1
If P=[3212−1232],A=[1101]P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] and Q=PAPTQ=P A P^T, then PTQ2005PP^T Q^{2005} P, is
View answer and explanation
Correct option: A
[1200501]\left[\begin{array}{cc}1 & 2005 \\ 0 & 1\end{array}\right]
Option A ([1200501]\left[\begin{array}{cc}1 & 2005 \\ 0 & 1\end{array}\right]) is correct because PTQ2005P=A2005P^T Q^{2005} P = A^{2005}, and An=[1n01]A^n = \left[\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right].
495 Easy
495 of 922
Application +4 | -1
The value of ∣log⁡3512log⁡43log⁡38log⁡49∣×∣log⁡23log⁡83log⁡34log⁡34∣\left| {\begin{array}{cccccccccccccccccccc} {{{\log }_3}512}&{{{\log }_4}3}\\ {{{\log }_3}8}&{{{\log }_4}9} \end{array}} \right| \times \left| {\begin{array}{cccccccccccccccccccc} {{{\log }_2}3}&{{{\log }_8}3}\\ {{{\log }_3}4}&{{{\log }_3}4} \end{array}} \right| is
View answer and explanation
Correct option: B
10
Option B (10) is correct because after simplifying the logarithmic expressions and evaluating the determinant, the result is 10.

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
496 Easy
496 of 922
Application +4 | -1
If the matrices A=[213410]A = \left[ {\begin{array}{cccccccccccccccccccc} 2&1&3\\ 4&1&0 \end{array}} \right] and B=[1−10250]B = \left[ {\begin{array}{cccccccccccccccccccc} 1&{ - 1}\\ 0&2\\ 5&0 \end{array}} \right], then ABAB
View answer and explanation
Correct option: A
[1704−2]\left[ {\begin{array}{cccccccccccccccccccc} {17}&0\\ 4&{ - 2} \end{array}} \right]
Option A ([17 0], [4 -2]) is correct because it is the result of the matrix multiplication A*B as shown in the solution.
497 Easy
497 of 922
Analyzing +4 | -1
If AA is a square matrix of order 2 such that A[1−1]=[−12]A\left[\begin{array}{r}1 \\ -1\end{array}\right]=\left[\begin{array}{r}-1 \\ 2\end{array}\right] and A2[1−1]=[10].A^2\left[\begin{array}{r}1 \\ -1\end{array}\right]=\left[\begin{array}{l}1 \\ 0\end{array}\right]. The sum of elements and product of elements of AA are SS and PP, then S+PS+P is
View answer and explanation
Correct option: D
5
Option D (5) is correct because by solving the matrix equations, we find the elements of matrix A, and then calculate S+P = 5.
498 Easy
498 of 922
Application +4 | -1
If
A=[3−41−1]A=\left[\begin{array}{ll} 3 & -4 \\ 1 & -1 \end{array}\right]
then ∣A2003∣−4∣A2002∣=\left|A^{2003}\right|-4\left|A^{2002}\right|=
View answer and explanation
Correct option: A
−3-3
Option A (-3) is correct because calculating the determinant of A and using the property ∣An∣=∣A∣n\left|A^{n}\right| = \left|A\right|^{n} simplifies the expression to -3.
499 Easy
499 of 922
Analyzing +4 | -1
If A=[aij]A=\left[a_{i j}\right] is a square matrix of even order such that aij=i2−j2a_{i j}=i^2-j^2, then
View answer and explanation
Correct option: D
none of these
The correct answer is none of these because A is a skew-symmetric matrix, and the determinant of a skew-symmetric matrix of even order is a perfect square, not necessarily zero.
500 Easy
500 of 922
Application +4 | -1
If A=diag⁡[321]\mathrm{A}=\operatorname{diag}\left[\begin{array}{lll}3 & 2 & 1\end{array}\right], then the sum of the elements of A−1=\mathrm{A}^{-1}=
View answer and explanation
Correct option: B
116\frac{11}{6}
Option D is correct. The adjugate of A is equal to 3 times the transpose of A.
501 Easy
501 of 922
Analyzing +4 | -1
Let the system of linear equations
4x+λy+2z=04x + \lambda y + 2z = 0
2x−y+z=02x - y + z = 0
μx+2y+3z=0,  λ,  μ∈R\mu x + 2y + 3z = 0,\,\,\lambda ,\,\,\mu \in R.
has a non-trivial solution. Then which of the following is true?
View answer and explanation
Correct option: A
μ=6, λ∈R\mu = 6,\,\lambda \in R
Option A (μ = 6, λ ∈ R) is correct because for a non-trivial solution, the determinant of the coefficient matrix must be zero, leading to this condition.
502 Easy
502 of 922
Understanding +4 | -1
If A\mathrm{A} and B\mathrm{B} are 2×22 \times 2 matrices, then which of the following is true ?
View answer and explanation
Correct option: C
(A−B)(A+B)=A2+AB−BA−B2(A-B)(A+B)=A^2+A B-B A-B^2
Option C ((A-B)(A+B)=A²+AB-BA-B²) is correct because matrix multiplication is not commutative, so AB is not necessarily equal to BA.
503 Easy
503 of 922
Analyzing +4 | -1
If PP is a 3×33 \times 3 real matrix such that PT=aP+(a−1)P^T=a P+(a-1) l, where a >1>1, then
View answer and explanation
Correct option: D
∣Adj⁡P∣=1|\operatorname{Adj} P|=1
Option D (∣Adj⁡P∣=1|\operatorname{Adj} P|=1) is correct because the given condition leads to P being the negative identity matrix, and thus ∣Adj⁡(P)∣=1|\operatorname{Adj}(P)| = 1.
504 Easy
504 of 922
Understanding +4 | -1
Let A,B{A}, {B} and C{C} are three matrices of same order. Now, consider the following statements

I. If A=B{A}={B}, then AC=BC{AC}={BC}

II. If AC=BC{AC}={BC}, then A=B{A}={B}

Choose the correct option
View answer and explanation
Correct option: A
Only II is true
Option A (Only I is true) is correct because if A=B, then AC=BC, but the converse is not always true.
505 Easy
505 of 922
Application +4 | -1
Let A=(4−2αβ)A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)

If A2+γA+18l=0A^2+\gamma A+18 l=0, then det⁡(A)\operatorname{det}(A) is equal to _____.
View answer and explanation
Correct option: B
18
Option B (18) is correct because by finding the characteristic equation and comparing it with the given equation, the determinant of A can be determined.
506 Easy
506 of 922
Application +4 | -1
If  A=[1011]  and  I=[1001],\mathrm{If\;}A = \left[ {\begin{array}{cccccccccccccccccccc} 1&0\\ 1&1 \end{array}} \right]\;\mathrm{and\;}I = \left[ {\begin{array}{cccccccccccccccccccc} 1&0\\ 0&1 \end{array}} \right], then which one of the following holds for all n≥1n \ge 1, by the principle of mathematical induction?
View answer and explanation
Correct option: D
An=nA−(n−1)I{A^n} = nA - \left( {n - 1} \right)I
Option D is correct because by induction, An=nA−(n−1)I{A^n} = nA - (n - 1)I holds true for all n >= 1.
507 Easy
507 of 922
Analyzing +4 | -1
If the system of linear equations x−2y+kz=1,2x+y+z=2,3x−y−kz=3,x - 2y + kz = 1,2x + y + z = 2,3x - y - kz = 3, has a solution (x,y,z),z≠0,\left( {x,y,z} \right),\mathrm{ }z \ne 0, then (x,y)(x,y) lies on the straight line whose equation is
View answer and explanation
Correct option: C
4x−3y−4=04x - 3y - 4 = 0
Option C (4x−3y−4=04x - 3y - 4 = 0) is correct because adding equations (i) and (iii) eliminates zz and directly yields the equation of the line on which (x,y)(x, y) lies.
508 Easy
508 of 922
Understanding +4 | -1
Let AA be a 3×33 \times 3 matrix such that A2−5A+7I=0{A^2} - 5A + 7I = 0
Statement- I A−1=17(5I−A)A^{-1}=\frac{1}{7}(5 I-A)
Statement - II The polynomial A3−2A2−3A+IA^3-2 A^2-3 A+I can be reduced to, 5(A−4I)5(A - 4I)
Then
View answer and explanation
Correct option: A
Statement - 1 is false, but statement - 2 is true
Option A (Statement - 1 is false, but statement - 2 is true) is correct because Statement I is actually true, and Statement II is also true.
509 Easy
509 of 922
Application +4 | -1
If A=[1tan⁡x−tan⁡x1]\mathrm{A}=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right], then the value of ∣A′A−1∣\left|\mathrm{A}^{\prime} \mathrm{A}^{-1}\right|
View answer and explanation
Correct option: B
4
Option C (1) is correct because ∣A′A−1∣=1\left|A^{\prime} A^{-1}\right| = 1 since the determinant of the product of a matrix and its inverse is always 1.
510 Easy
510 of 922
Analyzing +4 | -1
If α,β\alpha, \beta and γ\gamma are such that α+β+γ=0\alpha+\beta+\gamma=0, then ∣1cos⁡γcos⁡βcos⁡γ1cos⁡αcos⁡βcos⁡α1∣\left|\begin{array}{ccc}1 & \cos \gamma & \cos \beta \\ \cos \gamma & 1 & \cos \alpha \\ \cos \beta & \cos \alpha & 1\end{array}\right| is equal to
View answer and explanation
Correct option: D
none of these
Option A (Independent of α\alpha) is correct because the determinant simplifies to an expression that does not contain alpha.
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