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Beyond Advanced · Matrices and Determinants

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1
Let JJ be the 3×33\times3 matrix all of whose entries equal 11. Find the number of 3×33\times3 matrices XX, every entry of which is 00 or 11, that satisfy XJ=JXXJ=JX.
Not attempted
Your answer: —
Correct answer: 14
2
Let A=[123014001]A=\begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1\end{bmatrix}. Find the sum of all nine entries of the matrix A20−20A+19IA^{20}-20A+19I, where II is the 3×33\times3 identity matrix.
Not attempted
Your answer: —
Correct answer: 1520
3
Let AA and BB be square matrices of the same order with real entries satisfying A2=B2=IA^{2}=B^{2}=I and AB=−BAAB=-BA. Which of the following statements are correct?
Not attempted
A
For every such pair (of any order), (A+B)3=4(A+B)(A+B)^{3}=4(A+B).
B
For every such pair of 2×22\times2 real matrices, tr⁡A=0\operatorname{tr}A=0 and tr⁡B=0\operatorname{tr}B=0.
C
Such a pair A,BA,B exists for 3×33\times3 real matrices.
D
For every such pair (of any order), (A+B+AB)2=I(A+B+AB)^{2}=I.
4
The unit square with vertices O(0,0)O(0,0), P(1,0)P(1,0), R(1,1)R(1,1), Q(0,1)Q(0,1) is first transformed by the linear map whose matrix is A=[1201]A=\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}, and the resulting figure is then transformed by the linear map whose matrix is B=[1011]B=\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}. In each diagram the shaded parallelogram has vertices OO, P′P', R′R', Q′Q', where P′P' and Q′Q' are the final positions of the points PP and QQ. Which diagram shows the final figure?
Not attempted
A
β\beta
xyOP′Q′R′
B
α\alpha
xyOP′Q′R′
C
γ\gamma
xyOP′Q′R′
D
δ\delta
xyOP′Q′R′
5
Let P=[001100010]P=\begin{bmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix} and X=I+P−P2X=I+P-P^{2}, where II is the 3×33\times3 identity matrix. Find the entry in the first row and second column of X10X^{10}.
Not attempted
Your answer: —
Correct answer: 683
6
Let AA and BB be nilpotent matrices of the same order, that is, Ap=OA^{p}=O and Bq=OB^{q}=O for some positive integers p,qp,q. Which of the following statements are correct?
Not attempted
A
If AB=OAB=O (but BABA need not be OO), then A+BA+B is nilpotent.
B
A+BA+B is nilpotent for every such pair A,BA,B.
C
ABAB is nilpotent for every such pair A,BA,B.
D
If AB=BAAB=BA, then A+BA+B is nilpotent.
7
Real 2×22\times2 matrices A≠OA\neq O and BB satisfy AB−BA=AAB-BA=A. Which of the following must be true?
Not attempted
A
A2=AA^{2}=A
B
AA is invertible
C
AB=BAAB=BA
D
A2=OA^{2}=O
8
The triangle OPQOPQ with O(0,0)O(0,0), P(2,0)P(2,0), Q(0,1)Q(0,1) is mapped by a matrix MM to the triangle OP′Q′OP'Q', where P′P' and Q′Q' are the images of PP and QQ. List-I gives four 2×22\times2 matrices and List-II refers to the five numbered figures in the picture, each showing a triangle OP′Q′OP'Q' with P′P' and Q′Q' labelled. Match each matrix with the figure that shows the image of triangle OPQOPQ under it. List-IList-IIP) [1101]−21) figure 1Q) [0−110]32) figure 2R) [0110]20253) figure 3S) [1011][1101]4) figure 45) figure 5\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \begin{bmatrix}1&1\\0&1\end{bmatrix}^{-2} & \text{1)}\ \text{figure 1} \\ \text{Q)}\ \begin{bmatrix}0&-1\\1&0\end{bmatrix}^{3} & \text{2)}\ \text{figure 2} \\ \text{R)}\ \begin{bmatrix}0&1\\1&0\end{bmatrix}^{2025} & \text{3)}\ \text{figure 3} \\ \text{S)}\ \begin{bmatrix}1&0\\1&1\end{bmatrix}\begin{bmatrix}1&1\\0&1\end{bmatrix} & \text{4)}\ \text{figure 4} \\ & \text{5)}\ \text{figure 5} \end{array}
Not attempted
xyOP′Q′1xyOP′Q′2xyOP′Q′3xyOP′Q′4xyOP′Q′5
A
P-4, Q-2, R-5, S-1
B
P-1, Q-5, R-2, S-3
C
P-3, Q-5, R-2, S-1
D
P-3, Q-2, R-5, S-4
9
A directed graph has vertices 1,2,31,2,3 and exactly four arcs: 1→21\to2, 2→32\to3, 3→13\to1 and 3→23\to2 (shown in the figure; there are no other arcs and no loops). A walk of length nn is a sequence of nn consecutive arcs, and arcs may be reused. Find the number of walks of length 2020 that start at vertex 33 and end at vertex 22.
Not attempted
123
Your answer: —
Correct answer: 151
10
Let SS be a real symmetric 3×33\times3 matrix and KK a real skew-symmetric 3×33\times3 matrix. Which of the following statements are true for all such SS and KK?
Not attempted
A
det⁡(SK−KS)=0\det(SK-KS)=0.
B
SK−KSSK-KS is symmetric.
C
S2KS^{2}K is skew-symmetric.
D
tr⁡(SK)=0\operatorname{tr}(SK)=0.
11
Find the number of real skew-symmetric 3×33\times3 matrices AA, all of whose entries belong to {−1,0,1}\{-1,0,1\}, that satisfy A3+2A=OA^{3}+2A=O.
Not attempted
A
1313
B
1212
C
77
D
2424
12
Let A=[012−102−2−20]A=\begin{bmatrix}0 & 1 & 2 \\ -1 & 0 & 2 \\ -2 & -2 & 0\end{bmatrix} and Q=(I−A)(I+A)−1Q=(I-A)(I+A)^{-1}. Find tr⁡(Q)\operatorname{tr}(Q), correct to 22 decimal places.
Not attempted
Your answer: —
Correct answer: -0.6
13
Let AA be a real skew-symmetric 3×33\times3 matrix with tr⁡(A2)=−6\operatorname{tr}(A^{2})=-6. Then tr⁡(A4)\operatorname{tr}(A^{4}) equals
Not attempted
A
1818
B
3636
C
2727
D
1212
14
Find the number of 2×22\times2 matrices MM, with every entry in {−1,0,1}\{-1,0,1\}, that satisfy MMT=MTMMM^{T}=M^{T}M.
Not attempted
A
2727
B
3333
C
4545
D
3636
15
A directed graph on the vertices 1,2,31,2,3 has exactly the four arcs 1→21\to2, 2→12\to1, 1→31\to3 and 3→23\to2 (see the figure). Let A=[aij]A=[a_{ij}] be its adjacency matrix, with aij=1a_{ij}=1 if there is an arc i→ji\to j and aij=0a_{ij}=0 otherwise. Find det⁡(I+A−AT)\det\big(I+A-A^{T}\big).
Not attempted
123
Your answer: —
Correct answer: 3
16
Let K=[010−1010−10]K=\begin{bmatrix}0 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & -1 & 0\end{bmatrix}. In each diagram below, a 3×33\times3 grid has its shaded cells at the positions of the non-zero entries of one particular matrix (rows run top to bottom, columns left to right). Which diagram shows the non-zero entries of adj⁡K\operatorname{adj}K?
Not attempted
A
β\beta
B
γ\gamma
C
α\alpha
D
δ\delta
17
AA is a real 3×23\times2 matrix with ATA=[5332]A^{T}A=\begin{bmatrix}5 & 3 \\ 3 & 2\end{bmatrix}. Find tr⁡((AAT)3)\operatorname{tr}\big((AA^{T})^{3}\big).
Not attempted
Your answer: —
Correct answer: 322
18
Let M=[231−1]M=\begin{bmatrix}2 & 3 \\ 1 & -1\end{bmatrix} and let C=[Cij]C=[C_{ij}] be its matrix of cofactors, Cij=(−1)i+jMijC_{ij}=(-1)^{i+j}M_{ij} (not transposed). The unit square OPRQOPRQ with P(1,0)P(1,0), R(1,1)R(1,1), Q(0,1)Q(0,1) is mapped by the linear map whose matrix is CC. In each diagram the shaded parallelogram OP′R′Q′OP'R'Q' has P′P' and Q′Q' marked. Which diagram shows the image of the unit square under CC?
Not attempted
A
β\beta
xyOP′Q′R′
B
γ\gamma
xyOP′Q′R′
C
δ\delta
xyOP′Q′R′
D
α\alpha
xyOP′Q′R′
19
For a real number k≥0k\ge0, let V(k)=∣14kk144k1∣V(k)=\begin{vmatrix}1&4&k\\k&1&4\\4&k&1\end{vmatrix}. Find the minimum value of V(k)V(k) over all k≥0k\ge0.
Not attempted
A
4949
B
6565
C
8181
D
7373
20
Let M=[1234567810]M=\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10\end{bmatrix} and let CijC_{ij} denote the cofactor of the entry in row ii, column jj of MM. Note det⁡M=−3\det M=-3. Which of the following equalities are true?
Not attempted
A
4C11+5C12+6C13=−34C_{11}+5C_{12}+6C_{13}=-3
B
2C21+5C22+8C23=−32C_{21}+5C_{22}+8C_{23}=-3
C
7C11+8C12+10C13=−37C_{11}+8C_{12}+10C_{13}=-3
D
2C12+5C22+8C32=−32C_{12}+5C_{22}+8C_{32}=-3
21
The triangular region RR in the figure has vertices (−1,0)(-1,0), (6,0)(6,0) and (2,4)(2,4). Let A=[21−13]A=\begin{bmatrix}2 & 1 \\ -1 & 3\end{bmatrix} and B=[123−1]B=\begin{bmatrix}1 & 2 \\ 3 & -1\end{bmatrix}, and let MM be the 2×22\times2 matrix satisfying A M AT=2BA\,M\,A^{T}=2B. The map x⃗↦Mx⃗\vec x\mapsto M\vec x sends RR to a region R′R'. Find the area of R′R'.
Not attempted
xyO(−1, 0)(6, 0)(2, 4)R
A
44
B
1414
C
5656
D
88
22
Each entry of a 3×33\times3 matrix is independently chosen to be +1+1 or −1-1, giving 29=5122^9=512 matrices. How many of these matrices have non-zero determinant?
Not attempted
Your answer: —
Correct answer: 192
23
Let aija_{ij} (1≤i,j≤31\le i,j\le3) be real numbers with 0≤aij≤10\le a_{ij}\le1 for all i,ji,j. The maximum possible value of det⁡ [aij]3×3\det\,[a_{ij}]_{3\times3} is
Not attempted
A
11
B
22
C
33
D
44
24
Three vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c in space each have length 33, and the angle between any two of them is 60∘60^\circ. Find the volume of the parallelepiped determined by them, correct to 22 decimal places.
Not attempted
Your answer: —
Correct answer: 19.09
25
Let MM be a real 3×33\times3 matrix with det⁡M=−2\det M=-2. Which of the following statements about the stated linear maps of space are correct? (Volume factor means the factor by which the volume of every solid is multiplied.)
Not attempted
A
The map x⃗↦adj⁡(M) x⃗\vec x\mapsto\operatorname{adj}(M)\,\vec x multiplies volumes by 44 and preserves orientation.
B
The map x⃗↦(2M)−1x⃗\vec x\mapsto(2M)^{-1}\vec x sends a solid of volume 3232 to a solid of volume 22 and reverses orientation.
C
The map x⃗↦MTM2x⃗\vec x\mapsto M^{T}M^{2}\vec x reverses orientation.
D
The map x⃗↦(M−1adj⁡M) x⃗\vec x\mapsto(M^{-1}\operatorname{adj}M)\,\vec x multiplies volumes by 12\tfrac12.
26
The numbers a,b,ca,b,c are the roots of x3−6x2+9x−2=0x^{3}-6x^{2}+9x-2=0. Find the value of ∣a2b2c2b2c2a2c2a2b2∣\begin{vmatrix}a^{2}&b^{2}&c^{2}\\b^{2}&c^{2}&a^{2}\\c^{2}&a^{2}&b^{2}\end{vmatrix}.
Not attempted
Your answer: —
Correct answer: -2754
27
The real numbers a,b,ca,b,c are the three roots of x3−3x+1=0x^{3}-3x+1=0, whose graph is shown. Let D=∣1a2a31b2b31c2c3∣D=\begin{vmatrix}1&a^{2}&a^{3}\\1&b^{2}&b^{3}\\1&c^{2}&c^{3}\end{vmatrix} (the order of a,b,ca,b,c is not specified). Find ∣D∣|D|.
Not attempted
xyOy = x³ − 3x + 1
A
99
B
00
C
2727
D
8181
28
Let R1,R2,R3R_1,R_2,R_3 be the rows of a 3×33\times3 matrix AA with det⁡A=5\det A=5. For each statement, the determinant of the new matrix is formed from the listed rows in the stated order. Which statements are correct?
Not attempted
A
Rows R1+2R2, 2R2+3R3, 3R3+R1R_1+2R_2,\ 2R_2+3R_3,\ 3R_3+R_1 give determinant 3030.
B
Rows R2, R1, R3+4R1R_2,\ R_1,\ R_3+4R_1 give determinant −5-5.
C
Rows R1+R2, R2+R3, R3+R1R_1+R_2,\ R_2+R_3,\ R_3+R_1 give determinant 1010.
D
Rows R1−R2, R2−R3, R3−R1R_1-R_2,\ R_2-R_3,\ R_3-R_1 give determinant 55.
29
Let W(x)=∣xexe2x1ex2e2x0ex4e2x∣W(x)=\begin{vmatrix}x&e^{x}&e^{2x}\\1&e^{x}&2e^{2x}\\0&e^{x}&4e^{2x}\end{vmatrix}. Find W′′(0)W''(0), the second derivative of WW at x=0x=0.
Not attempted
A
−7-7
B
1515
C
−16-16
D
−15-15
30
For integers p,qp,q with −5≤p,q≤5-5\le p,q\le5, let Δ(p,q)=∣qpppqpppq∣\Delta(p,q)=\begin{vmatrix}q&p&p\\p&q&p\\p&p&q\end{vmatrix}. Find the number of ordered pairs (p,q)(p,q) for which Δ(p,q)=0\Delta(p,q)=0.
Not attempted
A
1515
B
1616
C
1111
D
2121
31
Let g(x)=∣x111x111x∣g(x)=\begin{vmatrix}x&1&1\\1&x&1\\1&1&x\end{vmatrix}. Which graph shows y=g(x)y=g(x)? (Every graph is drawn on the same axes with the same scale; the tick labels show xx-values.)
Not attempted
A
β\beta
xyO-3-2-1123
B
α\alpha
xyO-3-2-1123
C
γ\gamma
xyO-3-2-1123
D
δ\delta
xyO-3-2-1123
32
Match each determinant in List-I with its value in List-II (the letters a,b,ca,b,c are arbitrary real numbers). List-IList-IIP) ∣abcbcacab∣1) 4abcQ) ∣b+caabc+abcca+b∣2) (a−b)(b−c)(a−c)R) ∣1abc1bca1cab∣3) (a−b)(b−c)(c−a)S) ∣a2a1b2b1c2c1∣4) 3abc−a3−b3−c35) abc+ab+bc+ca\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} & \text{1)}\ 4abc \\ \text{Q)}\ \begin{vmatrix}b+c&a&a\\b&c+a&b\\c&c&a+b\end{vmatrix} & \text{2)}\ (a-b)(b-c)(a-c) \\ \text{R)}\ \begin{vmatrix}1&a&bc\\1&b&ca\\1&c&ab\end{vmatrix} & \text{3)}\ (a-b)(b-c)(c-a) \\ \text{S)}\ \begin{vmatrix}a^{2}&a&1\\b^{2}&b&1\\c^{2}&c&1\end{vmatrix} & \text{4)}\ 3abc-a^{3}-b^{3}-c^{3} \\ & \text{5)}\ abc+ab+bc+ca \end{array}
Not attempted
A
P-4, Q-1, R-2, S-3
B
P-5, Q-4, R-3, S-2
C
P-4, Q-1, R-3, S-2
D
P-4, Q-5, R-2, S-3
33
Let a1,a2,a3=0,1,3a_1,a_2,a_3=0,1,3 and b1,b2,b3=0,2,5b_1,b_2,b_3=0,2,5, and let EE be the 3×33\times3 matrix with entries Eij=(ai−bj)2E_{ij}=(a_i-b_j)^{2}. Find det⁡E\det E.
Not attempted
Your answer: —
Correct answer: 360
34
A 3×33\times3 real matrix AA is transformed into the identity matrix II by the following five elementary row operations, applied in this order: R1↔R3R_1\leftrightarrow R_3; then R2→R2−5R1R_2\to R_2-5R_1; then R1→12R1R_1\to\tfrac12R_1; then R3→R3+R2R_3\to R_3+R_2; then R2→14R2R_2\to\tfrac14R_2. Find the value of ∣adj⁡(2A)∣\big|\operatorname{adj}(2A)\big|, the determinant of the adjoint of 2A2A.
Not attempted
Your answer: —
Correct answer: 4096
35
Which of the following statements are true?
Not attempted
A
For every real 2×22\times2 matrix XX: adj⁡(adj⁡X)=X\operatorname{adj}(\operatorname{adj}X)=X.
B
If X,YX,Y are 3×33\times3 real matrices with XY=YXXY=YX, then adj⁡X⋅adj⁡Y=adj⁡Y⋅adj⁡X\operatorname{adj}X\cdot\operatorname{adj}Y=\operatorname{adj}Y\cdot\operatorname{adj}X.
C
For every real 3×33\times3 matrix XX and real kk: adj⁡(kX)=k adj⁡X\operatorname{adj}(kX)=k\,\operatorname{adj}X.
D
For all real 2×22\times2 matrices X,YX,Y: adj⁡(X+Y)=adj⁡X+adj⁡Y\operatorname{adj}(X+Y)=\operatorname{adj}X+\operatorname{adj}Y.
36
Let u=[123]u=\begin{bmatrix}1\\2\\3\end{bmatrix}, v=[1−12]v=\begin{bmatrix}1\\-1\\2\end{bmatrix} and A=I+uvTA=I+uv^{T}, where II is the 3×33\times3 identity matrix. Find the trace of adj⁡A\operatorname{adj}A.
Not attempted
A
1818
B
1111
C
55
D
1313
37
Let A=[130013301]A=\begin{bmatrix}1&3&0\\0&1&3\\3&0&1\end{bmatrix}. Find the sum of all nine entries of adj⁡A\operatorname{adj}A.
Not attempted
A
2121
B
8484
C
2828
D
77
38
Let AA be a real 2×32\times3 matrix and BB a real 3×23\times2 matrix with AB=I2AB=I_{2}. Which of the following must hold for the 3×33\times3 matrix BABA?
Not attempted
A
BA=I3BA=I_{3} and tr⁡(BA)=3\operatorname{tr}(BA)=3
B
(BA)2=BA, det⁡(BA)=1, tr⁡(BA)=3(BA)^{2}=BA,\ \det(BA)=1,\ \operatorname{tr}(BA)=3
C
(BA)2=BA, det⁡(BA)=0, tr⁡(BA)=2(BA)^{2}=BA,\ \det(BA)=0,\ \operatorname{tr}(BA)=2
D
(BA)2=I3, det⁡(BA)=0, tr⁡(BA)=2(BA)^{2}=I_{3},\ \det(BA)=0,\ \operatorname{tr}(BA)=2
39
Find the number of 2×22\times2 matrices XX, all of whose entries belong to {−1,0,1}\{-1,0,1\}, such that XX is invertible and every entry of X−1X^{-1} is also an integer.
Not attempted
A
88
B
4040
C
2020
D
3232
40
Let AA be a 3×33\times3 real matrix with ∣A∣=−2|A|=-2, where ∣X∣|X| denotes the determinant. Match each quantity in List-I with its value in List-II. List-IList-IIP) ∣adj⁡A∣1) −2Q) ∣adj⁡(adj⁡A)∣2) 2R) ∣adj⁡(2A)∣3) 4S) ∣A−1adj⁡(AT)∣4) 165) 256\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \big|\operatorname{adj}A\big| & \text{1)}\ -2 \\ \text{Q)}\ \big|\operatorname{adj}(\operatorname{adj}A)\big| & \text{2)}\ 2 \\ \text{R)}\ \big|\operatorname{adj}(2A)\big| & \text{3)}\ 4 \\ \text{S)}\ \big|A^{-1}\operatorname{adj}(A^{T})\big| & \text{4)}\ 16 \\ & \text{5)}\ 256 \end{array}
Not attempted
A
P-3, Q-4, R-2, S-5
B
P-2, Q-4, R-5, S-3
C
P-3, Q-2, R-5, S-1
D
P-3, Q-4, R-5, S-1
41
The triangle OP′Q′OP'Q' in the figure has O=(0,0)O=(0,0), P′=(3,1)P'=(3,1) and Q′=(2,4)Q'=(2,4). Let AA be the 2×22\times2 matrix whose first column is OP′→\overrightarrow{OP'} and whose second column is OQ′→\overrightarrow{OQ'}. The matrix MM satisfies MA=2adj⁡AM A=2\operatorname{adj}A. Find the area of the image of the triangle OP′Q′OP'Q' under the map x⃗↦Mx⃗\vec x\mapsto M\vec x.
Not attempted
xyOP′ (3, 1)Q′ (2, 4)
A
1010
B
2020
C
55
D
5050
42
Consider the system x+2y+3z=ax+2y+3z=a,  2x+3y+4z=b\ 2x+3y+4z=b,  8x+13y+18z=c\ 8x+13y+18z=c. The constants a,b,ca,b,c are chosen independently from the seven integers −3,−2,−1,0,1,2,3-3,-2,-1,0,1,2,3, giving 343343 systems. How many of these systems have at least one solution (x,y,z)(x,y,z)?
Not attempted
A
3737
B
4949
C
1717
D
2121
43
A system AX=BAX=B of three linear equations in three unknowns has Δ=det⁡A=0\Delta=\det A=0. Let Δ1,Δ2,Δ3\Delta_1,\Delta_2,\Delta_3 be the determinants obtained by replacing the first, second, third column of AA by BB. Which of the following statements are correct?
Not attempted
A
If Δ1=Δ2=Δ3=0\Delta_1=\Delta_2=\Delta_3=0 and the system is inconsistent, then AA has rank at most 11.
B
If Δ1=Δ2=Δ3=0\Delta_1=\Delta_2=\Delta_3=0 and AA has rank 22, then the system is consistent.
C
If (adj⁡A)B=O(\operatorname{adj}A)B=O, then the system is consistent.
D
If Δ1=Δ2=Δ3=0\Delta_1=\Delta_2=\Delta_3=0, then the system has infinitely many solutions.
44
The real numbers x,y,zx,y,z satisfy the three equations x+ty+t2z=t3x+ty+t^{2}z=t^{3} for t=2t=2, t=3t=3 and t=5t=5. Find x+y+zx+y+z.
Not attempted
A
99
B
−8-8
C
77
D
7171
45
For an integer λ\lambda let Aλ=[λ−511λ−52001]A_\lambda=\begin{bmatrix}\lambda&-5&1\\1&\lambda-5&2\\0&0&1\end{bmatrix}. Find the sum of all integers λ\lambda for which the system AλX=BA_\lambda X=B has an integer solution XX for every integer column vector BB.
Not attempted
A
1010
B
55
C
00
D
1515
46
For an integer kk, consider the system x+ky=1x+ky=1,  kx+y=k2\ kx+y=k^{2} in the unknowns x,yx,y. For which of the following values of kk does the system have at least one solution (x,y)(x,y) in which both xx and yy are integers?
Not attempted
A
k=−2k=-2
B
k=3k=3
C
k=−1k=-1
D
k=1k=1
47
The three planes x+y+z=1x+y+z=1,  x+2y+3z=2\ x+2y+3z=2 and 2x+3y+4z=52x+3y+4z=5 are all parallel to the direction (1,−2,1)(1,-2,1). Imagine looking along this direction; each plane then appears as a straight line in the plane of the picture. Which diagram (drawn schematically) shows how the three planes appear?
Not attempted
A
β\beta
B
γ\gamma
C
α\alpha
D
δ\delta
48
For each system in List-I (unknowns x,y,zx,y,z), let Δ\Delta be the determinant of the coefficient matrix and Δ1,Δ2,Δ3\Delta_1,\Delta_2,\Delta_3 the Cramer determinants. Match each system with the description in List-II that it satisfies. List-IList-IIP) x+y+z=3, x+2y+3z=6, 2x+3y+4z=91) Δ≠0Q) x+y+z=3, x+y+z=4, x+y+z=52) Δ=0, some Δi≠0R) x+y+z=3, x+2y+3z=6, 2x+3y+4z=103) Δ=Δ1=Δ2=Δ3=0, infinitely many solutionsS) x+y+z=3, x−y+z=1, x+2y−z=24) Δ=Δ1=Δ2=Δ3=0, no solution5) Δ≠0, no solution\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ x+y+z=3,\ x+2y+3z=6,\ 2x+3y+4z=9 & \text{1)}\ \Delta\neq0 \\ \text{Q)}\ x+y+z=3,\ x+y+z=4,\ x+y+z=5 & \text{2)}\ \Delta=0,\ \text{some }\Delta_i\neq0 \\ \text{R)}\ x+y+z=3,\ x+2y+3z=6,\ 2x+3y+4z=10 & \text{3)}\ \Delta=\Delta_1=\Delta_2=\Delta_3=0,\ \text{infinitely many solutions} \\ \text{S)}\ x+y+z=3,\ x-y+z=1,\ x+2y-z=2 & \text{4)}\ \Delta=\Delta_1=\Delta_2=\Delta_3=0,\ \text{no solution} \\ & \text{5)}\ \Delta\neq0,\ \text{no solution} \end{array}
Not attempted
A
P-4, Q-3, R-2, S-1
B
P-3, Q-4, R-2, S-1
C
P-3, Q-2, R-4, S-1
D
P-2, Q-4, R-3, S-5
49
A real 3×33\times3 matrix AA and a column vector BB are such that AX=BAX=B has each of X1=(1,0,0)TX_1=(1,0,0)^T, X2=(0,1,0)TX_2=(0,1,0)^T and X3=(0,0,1)TX_3=(0,0,1)^T as a solution, and nothing else is known about AA or BB (in particular BB may or may not be zero). Which of the following vectors must also satisfy AX=BAX=B?
Not attempted
A
(1,1,1)T(1,1,1)^{T}
B
(2,2,−1)T(2,2,-1)^{T}
C
(12,12,12)T\left(\tfrac12,\tfrac12,\tfrac12\right)^{T}
D
(2,−3,2)T(2,-3,2)^{T}
50
Consider the three lines kx+2y=3kx+2y=3,  2x+ky=3\ 2x+ky=3 and x+y=2x+y=2 for a real number kk. Which of the following statements are correct?
Not attempted
A
For k=2k=2 the three lines pass through one point.
B
The determinant ∣k2−32k−311−2∣\begin{vmatrix}k&2&-3\\2&k&-3\\1&1&-2\end{vmatrix} vanishes for exactly two real values of kk.
C
If ∣k2−32k−311−2∣=0\begin{vmatrix}k&2&-3\\2&k&-3\\1&1&-2\end{vmatrix}=0, then the three lines have a common point.
D
For k=1k=1 the three lines pass through one point.
51
For each real number tt let PtP_t be the point (t, t2, t3)(t,\ t^{2},\ t^{3}) in space. For how many real values of λ\lambda are the four points P1, P2, P3, PλP_1,\ P_2,\ P_3,\ P_\lambda coplanar? (Points that coincide are allowed, and a set containing a repeated point counts as coplanar.)
Not attempted
Your answer: —
Correct answer: 3
52
The real numbers a=12a=\tfrac12 and b=13b=\tfrac13 are fixed, and cc is a real number such that the equations x=cy+bzx=cy+bz,  y=az+cx\ y=az+cx,  z=bx+ay\ z=bx+ay have a solution (x,y,z)≠(0,0,0)(x,y,z)\neq(0,0,0). Find the sum of the squares of all possible values of cc, correct to 22 decimal places.
Not attempted
Your answer: —
Correct answer: 1.39
53
The real numbers x,y,zx,y,z are positive and satisfy x=12(y+z)x=\tfrac12(y+z),  y=13(z+x)\ y=\tfrac13(z+x),  z=c (x+y)\ z=c\,(x+y) for some real number cc. If also x2+y2+z2=50x^{2}+y^{2}+z^{2}=50, find x+y+zx+y+z.
Not attempted
Your answer: —
Correct answer: 12
54
Which of the following statements are correct?
Not attempted
A
For K=[03−2−3012−10]K=\begin{bmatrix}0&3&-2\\-3&0&1\\2&-1&0\end{bmatrix}, every solution of KX=OKX=O is a scalar multiple of (1,2,3)T(1,2,3)^{T}.
B
If X1X_1 and X2X_2 are solutions of AX=BAX=B with B≠OB\ne O, then X1+X2X_1+X_2 is also a solution of AX=BAX=B.
C
If a square system AX=OAX=O has a non-trivial solution, then AX=BAX=B has a solution for every column BB of the right size.
D
Every real skew-symmetric 3×33\times3 matrix KK has a non-zero column vector XX with KX=OKX=O.
55
The lines L1: x+y=2L_1:\ x+y=2 and L2: 2x−y=1L_2:\ 2x-y=1 are given, and for each real number kk let L3: kx+(k+1)y=2k+1L_3:\ kx+(k+1)y=2k+1. For how many real values of kk do L1L_1, L2L_2, L3L_3 have a point in common?
Not attempted
A
exactly one value of kk
B
all real values of kk
C
exactly two values of kk
D
no real value of kk
56
For a real angle θ\theta consider the system xcos⁡θ+ysin⁡θ+z=0x\cos\theta+y\sin\theta+z=0,  xcos⁡2θ+ysin⁡2θ+z=0\ x\cos2\theta+y\sin2\theta+z=0,  xcos⁡3θ+ysin⁡3θ+z=0\ x\cos3\theta+y\sin3\theta+z=0. The figure shows the points Pt=(cos⁡tθ,sin⁡tθ)P_t=(\cos t\theta,\sin t\theta), t=1,2,3t=1,2,3, on the unit circle for one value of θ\theta. Find the number of values of θ\theta in the interval [0,2π)[0,2\pi) for which the system has a solution (x,y,z)≠(0,0,0)(x,y,z)\neq(0,0,0).
Not attempted
xyOP1P2P3
A
22
B
33
C
44
D
11
57
Let A=[1−331]A=\begin{bmatrix}1&-\sqrt3\\\sqrt3&1\end{bmatrix}. Then A2025A^{2025} equals
Not attempted
A
22025 I2^{2025}\,I
B
22024 A2^{2024}\,A
C
−22024 A-2^{2024}\,A
D
−22025 I-2^{2025}\,I
58
Let A=[3−111]A=\begin{bmatrix}3&-1\\1&1\end{bmatrix}. Find the entry in the second row and first column of A10A^{10}.
Not attempted
Your answer: —
Correct answer: 5120
59
Let u=[123]u=\begin{bmatrix}1\\2\\3\end{bmatrix}, v=[2−11]v=\begin{bmatrix}2\\-1\\1\end{bmatrix} and A=uvTA=uv^{T} (a 3×33\times3 matrix). Which of the following statements are correct?
Not attempted
A
A−3IA-3I is a singular matrix.
B
I−AI-A is invertible and (I−A)−1=I−12A(I-A)^{-1}=I-\tfrac12A.
C
det⁡(I+A)=4\det(I+A)=4.
D
A10=310AA^{10}=3^{10}A.
60
A 2×22\times2 real matrix AA satisfies A2−3A+2I=OA^{2}-3A+2I=O, and AA is neither II nor 2I2I. Find det⁡(A+I)\det(A+I).
Not attempted
A
33
B
55
C
66
D
22
61
The triangle OPQOPQ with O(0,0)O(0,0), P(2,0)P(2,0), Q(0,1)Q(0,1) is mapped by a matrix MM to the triangle OP′Q′OP'Q', where P′P' and Q′Q' are the images of PP and QQ. List-I gives four powers of 2×22\times2 matrices and List-II refers to the five numbered figures in the picture, each showing a triangle OP′Q′OP'Q' with P′P' and Q′Q' labelled. Match each power with the figure that shows the image of triangle OPQOPQ under it. List-IList-IIP) [12−323212]1001) figure 1Q) [0−110]20262) figure 2R) [0110]20253) figure 3S) [−12−3232−12]404) figure 45) figure 5\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \begin{bmatrix}\tfrac12&-\tfrac{\sqrt3}2\\\tfrac{\sqrt3}2&\tfrac12\end{bmatrix}^{100} & \text{1)}\ \text{figure 1} \\ \text{Q)}\ \begin{bmatrix}0&-1\\1&0\end{bmatrix}^{2026} & \text{2)}\ \text{figure 2} \\ \text{R)}\ \begin{bmatrix}0&1\\1&0\end{bmatrix}^{2025} & \text{3)}\ \text{figure 3} \\ \text{S)}\ \begin{bmatrix}-\tfrac12&-\tfrac{\sqrt3}2\\\tfrac{\sqrt3}2&-\tfrac12\end{bmatrix}^{40} & \text{4)}\ \text{figure 4} \\ & \text{5)}\ \text{figure 5} \end{array}
Not attempted
xyOP′Q′1xyOP′Q′2xyOP′Q′3xyOP′Q′4xyOP′Q′5
A
P-5, Q-2, R-3, S-1
B
P-1, Q-3, R-2, S-4
C
P-4, Q-2, R-5, S-1
D
P-1, Q-2, R-3, S-4
62
A weighted directed graph has vertices 1,2,31,2,3 and exactly three arcs: 1→21\to2 of weight 11, 2→32\to3 of weight 33 and 1→31\to3 of weight 22 (see the figure). Let N=[nij]N=[n_{ij}] be the weight matrix, with nijn_{ij} the weight of the arc i→ji\to j and 00 if there is no such arc, and let A=I+NA=I+N. Find the sum of all nine entries of A−5A^{-5}.
Not attempted
123132
Your answer: —
Correct answer: 18
63
Find the number of 3×33\times3 matrices AA, every entry of which belongs to {−1,0,1}\{-1,0,1\}, that satisfy both ATA=IA^{T}A=I and A2=IA^{2}=I.
Not attempted
A
4848
B
2020
C
3232
D
88
64
Let P=[2−21−1]P=\begin{bmatrix}2 & -2 \\ 1 & -1\end{bmatrix}. In the figure, LL is the line y=x2y=\tfrac x2 and the vector v=(3,1)Tv=(3,1)^T is sent to Pv=(4,2)TPv=(4,2)^T on LL; the dashed segment joins the tip of vv to the tip of PvPv. Which of the following statements are correct?
Not attempted
xyOv (3, 1)Pv (4, 2)L
A
tr⁡P=1\operatorname{tr}P=1 and det⁡P=0\det P=0.
B
det⁡(2P−I)=−1\det(2P-I)=-1.
C
I−PI-P is idempotent and (I−P)P=O(I-P)P=O.
D
PP is the orthogonal projection of the plane onto LL.
65
Find the number of non-zero 2×22\times2 matrices [abcd]\begin{bmatrix}a&b\\c&d\end{bmatrix} with integer entries, each between −3-3 and 33 inclusive, that are nilpotent (that is, Mk=OM^{k}=O for some positive integer kk).
Not attempted
Your answer: —
Correct answer: 24
66
Let AA be a 2×22\times2 real orthogonal matrix with det⁡A=−1\det A=-1 whose first column is (35,45)T\left(\tfrac35,\tfrac45\right)^{T} (the figure shows the unit vector e1e_1 and its image Ae1Ae_1). Find the (1,2)(1,2) entry of S=A+A2+A3+⋯+A2025S=A+A^{2}+A^{3}+\cdots+A^{2025}, correct to 22 decimal places.
Not attempted
xyOe₁Ae₁
Your answer: —
Correct answer: 810.4
67
The triangle with vertices P(1,0)P(1,0), Q(3,0)Q(3,0), R(1,2)R(1,2) is mapped by a 2×22\times2 real orthogonal matrix AA with det⁡A=−1\det A=-1 and A[125]=[130]A\begin{bmatrix}12\\5\end{bmatrix}=\begin{bmatrix}13\\0\end{bmatrix}. In each diagram the shaded triangle has vertices P′,Q′,R′P',Q',R', the images of P,Q,RP,Q,R. Which diagram shows the image triangle?
Not attempted
A
β\beta
xyP′Q′R′
B
γ\gamma
xyP′Q′R′
C
α\alpha
xyP′Q′R′
D
δ\delta
xyP′Q′R′
68
Let RR be the 2×22\times2 matrix of the rotation through 120∘120^\circ about the origin (counterclockwise), as suggested by the figure, and let M(a,b,c)=aI+bR+cR2M(a,b,c)=aI+bR+cR^{2}. For how many triples of integers (a,b,c)(a,b,c) with −2≤a,b,c≤2-2\le a,b,c\le2 is M(a,b,c)M(a,b,c) a singular matrix?
Not attempted
xyOe₁Re₁R²e₁
A
55
B
11
C
2525
D
4545
69
A complex 3×33\times3 matrix AA satisfies Aθ=−AA^{\theta}=-A, where AθA^{\theta} denotes the conjugate transpose A‾ T\overline{A}^{\,T} (so AA is skew-Hermitian). Which of the following statements are true for every such AA?
Not attempted
A
det⁡A\det A is a real number.
B
iAiA is skew-Hermitian.
C
A2A^{2} is skew-Hermitian.
D
tr⁡(A2)\operatorname{tr}(A^{2}) is a real number that is at most 00.
70
Let AA be a real 2×32\times3 matrix and BB a real 3×23\times2 matrix with AB=[1234]AB=\begin{bmatrix}1&2\\3&4\end{bmatrix}. Nothing else is known about AA and BB. Match each quantity in List-I (the matrix BABA is 3×33\times3) with its value in List-II. List-IList-IIP) tr⁡(BA)1) 0Q) tr⁡((BA)2)2) 5R) tr⁡((BA)3)3) 29S) det⁡(BA)4) 1555) −2\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)}\ \operatorname{tr}(BA) & \text{1)}\ 0 \\ \text{Q)}\ \operatorname{tr}\big((BA)^{2}\big) & \text{2)}\ 5 \\ \text{R)}\ \operatorname{tr}\big((BA)^{3}\big) & \text{3)}\ 29 \\ \text{S)}\ \det(BA) & \text{4)}\ 155 \\ & \text{5)}\ -2 \end{array}
Not attempted
A
P-3, Q-2, R-4, S-1
B
P-2, Q-3, R-1, S-4
C
P-2, Q-3, R-4, S-1
D
P-2, Q-4, R-3, S-5
Write (BA)k=B(AB)k−1A(BA)^k=B(AB)^{k-1}A and use tr⁡(XY)=tr⁡(YX)\operatorname{tr}(XY)=\operatorname{tr}(YX) for non-square factors. For the determinant, compare the rank of BABA with its order.
Trace. (BA)k=B (AB)k−1A(BA)^k=B\,(AB)^{k-1}A, so tr⁡((BA)k)=tr⁡((AB)k−1AB)=tr⁡((AB)k)\operatorname{tr}\big((BA)^k\big)=\operatorname{tr}\big((AB)^{k-1}AB\big)=\operatorname{tr}\big((AB)^k\big) by tr⁡(XY)=tr⁡(YX)\operatorname{tr}(XY)=\operatorname{tr}(YX) (valid when both products exist). P: tr⁡(AB)=1+4=5\operatorname{tr}(AB)=1+4=5. Q: (AB)2=[7101522](AB)^2=\begin{bmatrix}7&10\\15&22\end{bmatrix} has trace 2929. R: (AB)3=[375481118](AB)^3=\begin{bmatrix}37&54\\81&118\end{bmatrix} has trace 155155. S: BABA is a product through a 22-dimensional space, so its rank is at most 22 and the 3×33\times3 matrix is singular: det⁡(BA)=0\det(BA)=0. The value −2=det⁡(AB)-2=\det(AB) is the trap for this row. Each wrong mapping differs from the correct one in at least two rows, for instance by interchanging two of the three traces or by using det⁡(AB)\det(AB) for det⁡(BA)\det(BA).