8 If  x, y, z are nonzero real numbers, then the inverse of matrix 
0
0
A
0
0
0
0
x
y
z




= 





is
(A)
1
1
1
0
0
0
0
0
0
x
y
z
−
−
−










(B)
1
1
1
0
0
0
0
0
0
x
xyz
y
z
−
−
−










(C)
0
0
1
0
0
0
0
x
y
xyz
z










(D)
1
0
0
1
0
1
0
0
0
1
xyz










9 Let A = 
1
sin
1
sin
1
sin
1
sin
1
θ




−
θ
θ




−
−
θ


, where 0 ≤ θ ≤ 2π Then
(A) Det (A) = 0
(B) Det (A) ∈ (2, ∞)
(C) Det (A) ∈ (2, 4)
(D) Det (A) ∈ [2, 4]
Rationalised 2023-24
DETERMINANTS     101
Summary
® Determinant of a matrix A = [a11]1×1 is given by |a11| = a11
® Determinant of a matrix A = 



a
a
a
a
11
12
21
22
 is given by
11
12
21
22
A
a
a
a
a
=
= a11 a22 – a12 a21
® Determinant of a matrix A =








a
b
c
a
b
c
a
b
c
1
1
1
2
2
2
3
3
3
 is given by (expanding along R1)
1
1
1
2
2
2
2
2
2
2
2
2
1
1
1
3
3
3
3
3
3
3
3
3
A
a
b
c
b
c
a
c
a
b
a
b
c
a
b
c
b
c
a
c
a
b
a
b
c
=
=
−
+
For any square matrix A, the |A| satisfy following properties