ECAT Mathematics
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ECAT Mathematics
Chapter 3: Matrices and Determinants
86 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
Square matrix has:
A
More rows than columns
B
Equal rows and columns
C
More columns than rows
D
No diagonal
1
Order 3×4 means:
A
4 rows 3 columns
B
3 rows 4 columns
C
12 elements in 1 row
D
7 elements
2
Transpose Aᵀ: swap
A
Elements with −1
B
Rows and columns
C
Diagonal elements
D
Nothing
3
Symmetric matrix: A=
A
−Aᵀ
B
Aᵀ
C
A⁻¹
D
0
4
Skew-symmetric: A=
A
Aᵀ
B
−Aᵀ
C
I
D
0
5
Identity matrix I satisfies:
A
AI=IA=0
B
AI=IA=A
C
A+I=A
D
det(I)=0
6
det[[1,2],[3,4]]=
A
−2
B
2
C
10
D
−10
7
det(A)=0: A is:
A
Non-singular
B
Singular
C
Identity
D
Diagonal
8
A⁻¹ exists iff:
A
A is square
B
det(A)≠0
C
A is symmetric
D
A is diagonal
9
For A=[[a,b],[c,d]]: A⁻¹=
A
[[d,−b],[−c,a]]/det
B
[[d,−b],[−c,a]]×det
C
[[a,c],[b,d]]/det
D
[[d,b],[c,a]]/det
10
Trace = sum of:
A
All elements
B
Diagonal elements
C
Off-diagonal elements
D
Column elements
11
Cramer's rule solves:
A
Matrix multiplication
B
System of linear equations
C
Eigenvalues
D
Transpositions
12
(AB)ᵀ=
A
AᵀBᵀ
B
BᵀAᵀ
C
AB
D
BA
13
Diagonal matrix: non-zero only on:
A
All positions
B
Main diagonal
C
Off-diagonal
D
Last column
14
Null (zero) matrix: all elements are:
A
1
B
0
C
Equal to diagonal
D
Different
15
det=ad−bc for [[a,b],[c,d]] is:
A
Minor
B
Cofactor
C
Determinant
D
Trace
16
Two identical rows: det=
A
1
B
0
C
2
D
−1
17
Swapping two rows:
A
Keeps det same
B
Doubles det
C
Changes sign of det
D
Makes det 0
18
Multiplying one row by k: det multiplied by:
A
1/k
B
k²
C
k
D
0
19
adj(A) = transpose of:
A
Minor matrix
B
Cofactor matrix
C
Identity matrix
D
Original matrix
20
A⁻¹=
A
det(A)/adj(A)
B
adj(A)/det(A)
C
adj(A)×det(A)
D
det(A)×A
21
AA⁻¹=
A
Zero matrix
B
A
C
Identity matrix
D
Aᵀ
22
(m×n)(n×p)=
A
m×p
B
n×n
C
m×n
D
p×m
23
Matrix multiplication:
A
Always commutative
B
Not commutative generally
C
AB=BA always
D
Not associative
24
Minor M_{ij} = det of matrix with row i and column j:
A
Removed
B
Doubled
C
Swapped
D
Added
25
Cofactor C_{ij}=
A
M_{ij}
B
(−1)^{i+j}M_{ij}
C
M_{ij}/det
D
(−1)ⁱM_{ij}
26
det(kA) for n×n:
A
k det(A)
B
kⁿ det(A)
C
det(A)
D
k² det(A)
27
Orthogonal matrix: AAᵀ=
A
0
B
A
C
I
D
Aᵀ
28
det(Aᵀ)=
A
det(−A)
B
−det(A)
C
det(A)
D
1/det(A)
29
det(AB)=
A
det(A)+det(B)
B
det(A)×det(B)
C
det(A)−det(B)
D
det(A)/det(B)
30
Rank of matrix = max number of linearly independent:
A
Determinants
B
Rows (in row echelon form)
C
Diagonal elements
D
Columns squared
31
Upper triangular: zeros are:
A
Above diagonal
B
Below diagonal
C
On diagonal
D
Everywhere
32
det of triangular matrix = product of:
A
All elements
B
Diagonal elements
C
Row sums
D
Column sums
33
Cofactor expansion of 3×3 can be along:
A
First row only
B
Any row or column
C
Main diagonal only
D
Last column only
34
Unique solution to AX=B when:
A
det(A)=0
B
det(A)≠0
C
A is symmetric
D
B=0
35
Characteristic polynomial: det(A−λI)=0 finds:
A
Eigenvectors
B
Eigenvalues
C
Trace
D
Rank
36
AX=λX defines:
A
Determinant
B
Eigenvalue equation
C
Trace
D
Transpose
37
det[[2,0],[0,3]]=
A
5
B
6
C
1
D
0
38
[[1,0,0],[0,1,0],[0,0,1]] is:
A
Null matrix
B
Identity matrix
C
Diagonal matrix
D
Scalar matrix
39
Element in row 2, column 3: notation
A
a_{23}
B
a_{32}
C
a_{2,3}
D
Both a and c
40
Gaussian elimination is used for:
A
Eigenvalues
B
Solving linear systems
C
Finding trace
D
Transposing
41
A²=A means A is:
A
Nilpotent
B
Idempotent
C
Involutory
D
Singular
42
A²=I means A is:
A
Nilpotent
B
Idempotent
C
Involutory
D
Singular
43
Aⁿ=0 for some n>0: A is:
A
Nilpotent
B
Idempotent
C
Symmetric
D
Orthogonal
44
System has no solution when:
A
det(A)≠0
B
det(A)=0 and inconsistent
C
A is symmetric
D
B=0
45
Rows of AB where A is 3×4, B is 4×2:
A
4
B
2
C
3
D
8
46
Matrix addition requires:
A
Same determinant
B
Same order (dimensions)
C
Same elements
D
One being square
47
Trace of [[3,1],[2,4]]=
A
7
B
5
C
10
D
12
48
det(I₃×₃)=
A
0
B
3
C
1
D
−1
49
Cofactor C_{11} of [[2,3],[4,5]]=
A
5
B
−5
C
3
D
−3
50
det(2A) for 2×2 with det(A)=5:
A
10
B
20
C
25
D
40
51
Cramer's rule: x = D_x/D where D_x:
A
det of A
B
det with x-column replaced by constants
C
det with y-column replaced
D
det of constant matrix
52
(A+B)ᵀ=
A
Aᵀ+Bᵀ
B
Aᵀ−Bᵀ
C
A+B
D
Bᵀ+A
53
Scalar matrix: diagonal elements are:
A
All equal to each other
B
All equal to 1
C
All different
D
Zero
54
Singular A: A⁻¹:
A
Exists
B
Does not exist
C
Equals A
D
Equals I
55
Non-singular A: AX=0 has:
A
Infinite solutions
B
No solution
C
Exactly X=0
D
Two solutions
56
A=[[4,7],[2,6]]: A⁻¹=
A
[[6,−7],[−2,4]]/10
B
[[6,−7],[−2,4]]/14
C
A itself
D
Undefined
57
REF: each non-zero row has:
A
All elements 1
B
Leading 1 with zeros below
C
Zeros everywhere
D
Equal elements
58
[[0,1],[1,0]] is a:
A
Identity matrix
B
Permutation matrix
C
Scalar matrix
D
Zero matrix
59
(A⁻¹)ᵀ=
A
(Aᵀ)⁻¹
B
A
C
A⁻¹
D
Aᵀ
60
Orthogonal matrix: det=
A
0
B
±1
C
2
D
Any real
61
Matrix addition: aᵢⱼ+bᵢⱼ gives:
A
Product
B
Sum of corresponding elements
C
Row sums
D
Column products
62
A=[[1,2],[3,4]], B=[[2,0],[1,3]]: AB=
A
[[4,6],[10,12]]
B
[[2,0],[3,12]]
C
[[4,6],[6,12]]
D
[[2,6],[10,4]]
63
Matrix multiplication is:
A
Commutative
B
Associative
C
Not associative
D
Only for square matrices
64
det(Aⁿ)=
A
n det(A)
B
[det(A)]ⁿ
C
n² det(A)
D
det(A)+n
65
Row operations preserve:
A
det(A)
B
Rank
C
Diagonal
D
Both det and rank
66
Cayley-Hamilton: A satisfies its own:
A
Row equation
B
Characteristic equation
C
Eigenvalue
D
Column equation
67
det[[0,0,1],[0,1,0],[1,0,0]]=
A
0
B
1
C
−1
D
2
68
Equal columns → det=
A
1
B
−1
C
0
D
2
69
Main diagonal: elements a_{ij} where:
A
i>j
B
i=j
C
i<j
D
i+j=n
70
(AB)⁻¹=
A
A⁻¹B⁻¹
B
B⁻¹A⁻¹
C
A⁻¹+B⁻¹
D
BA⁻¹
71
Lower triangular: zeros are:
A
Below diagonal
B
Above diagonal
C
On diagonal
D
Everywhere
72
n×n matrix has entries:
A
n
B
2n
C
n²
D
n³
73
det(2I₂×₂)=
A
2
B
4
C
0
D
1
74
[[1,0],[0,−1]] represents:
A
Reflection in x-axis
B
Reflection in y-axis
C
Identity
D
Zero matrix
75
Skew-symmetric: diagonal elements are:
A
1
B
0
C
Equal
D
Positive
76
Elementary row operations preserve:
A
det
B
Rank of matrix
C
Elements
D
Both
77
RREF: leading 1s with zeros:
A
Only below
B
Above and below
C
Only above
D
Nowhere
78
Coefficient matrix for x+y=3, 2x−y=1:
A
[[1,1],[2,−1]]
B
[[3],[1]]
C
[[1,3],[2,1]]
D
[[1,2],[1,−1]]
79
det(A+B) generally:
A
=det(A)+det(B)
B
≠det(A)+det(B)
C
=0
D
=det(A)det(B)
80
3×3 det expansion along row 1: a(ei−fh)−b(di−fg)+c(dh−eg) for [[a,b,c],[d,e,f],[g,h,i]]:
A
Correct
B
Wrong sign
C
Missing terms
D
Incomplete
81
Cramer's rule needs det(A):
A
=0
B
≠0
C
=1
D
<0
82
A×I=
A
Aᵀ
B
A⁻¹
C
A
D
0
83
det(kA) for n×n: =
A
k det(A)
B
det(A)
C
kⁿ det(A)
D
n det(A)
84
AX=0 with non-singular A: solution:
A
Infinite
B
No solution
C
X=0 only
D
X=A
85
A=[[cosθ,−sinθ],[sinθ,cosθ]]: det(A)=
A
0
B
−1
C
1
D
cos²θ
0
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