Notes and Study Materials
- Concepts of Determinants
- Master File for Determinants
- Master File for Adjoint and Inverse Matrices
- Determinants Note
- NCERT Solutions for – Determinants
- NCERT Exemplar Solutions for – Determinants
- R D Sharma Solution of Algebra of Determinants
- R D Sharma Solution of Linear Equation
- Past Many Years CBSE Questions and Answer Of Relation and Function
- Determinants Mind Map
Examples and Exercise
Determinants Notes Class 12 Maths Chapter 4
1. DETERMINANT OF A SQUARE MATRIX
(i) If A = \(\left[\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right]\), then det. A = \(\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right|\) = a11a22</sub – a21a12</sub
(ii) If A = \(\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]\), then det. A = a11\(\left|\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right|\) – a12 \(\left|\begin{array}{ll}
a_{21} & a_{23} \\
a_{31} & a_{33}
\end{array}\right|\) + a13 \(\left|\begin{array}{ll}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}\right|\)
= a11a22a33 – a23a32a11 – a12a21a33 + a12a23a11 + a13a21a32 – a13a31a22.
2. MINOR AND CO-FACTOR
(i) The minor of an element aij is a determinant, which is obtained by supressing die ith row and jth column. The minor of an element aij is denoted by Mij.
(ii) The co-factor of an element is its minor with proper sign. The co-factor of an element aij is denoted by Aij
Aij =(-1)i+jMij
3. PROPERTIES
(i) Reflection Property. The value of the determinant remains unaltered by interchanging its rows and columns.
(ii) Switching Property. If two adjacent rows (or columns) of a determinant are interchanged, then the sign of the determinant is changed.
(iii) Repetition Property. If two rows (or columns) of a determinant are identical, then its value is zero.
(iv) Scalar Multiple Property. If each element of a row (or column) of a determinant is multiplied f
by a constant ‘k’ then its value gets multiplied by the scalar ‘k’
(v) Sum Property. If each element of a row (or column) of a determinant is expressed as the sum
of two or more terms, then the determinant can be expressed as the sum of two or more determinants.
(vi) Invariance Property. If to any row (or column) of a determinant, a multiple of another row (or column) is added, the value of the determinant remains the same.
(vii) Factor Property. If a determinant Δ vanishes when for x is put a in those elements of Δ, which are polynomials in x, then (x – a) is a factor of Δ.
4. AREA OF A TRIANGLE
Area of a triangle whose vertices are (x1, y1), (x2, y2), (x3, y3) is given by:
D = \(\frac{1}{2}\left|\begin{array}{lll}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{array}\right|\)
When the area of the triangle is zero, then the points are collinear.
5. ADJOINT OF A MATRIX
Let A = \(\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]\), then adj, A = \(\left[\begin{array}{lll}
\mathrm{A}_{11} & \mathrm{~A}_{21} & \mathrm{~A}_{31} \\
\mathrm{~A}_{12} & \mathrm{~A}_{22} & \mathrm{~A}_{32} \\
\mathrm{~A}_{13} & \mathrm{~A}_{23} & \mathrm{~A}_{33}
\end{array}\right]\), where capital letters are co-factors of corresponding small letters.
6. INVERSE OF A MATRIX
Invertible Matrix. Any n-rowed square matrix A is said to be invertible if there exists an n-rowed matrix B such that
AB = BA = In
B is called the inverse of A and is denoted as A-1.
Theorems.
(i) Inverse of every square matrix, if it exists, is unique.
(ii) A is invertible iff |A| ≠ 0
(iii) A-1 = \(\frac{\operatorname{adj} . \mathrm{A}}{|\mathrm{A}|}\), if | A | ≠ 0.
PROPERTIES:
(i) (AB)-1 =B-1 A-1
(ii) (A’)-1 = (A-1)’
(iii) (Ak)-1 =(A-1)k, where k is any positive integer.
7. SINGULAR AND NON-SINGULAR MATRICES
A square matrix is said to be singular if |A| = 0 and non-singular if |A| ≠ 0.
8. SOLUTIONS OF EQUATIONS BY MATRIX METHOD To solve the equations :
\(\begin{array}{l}
a_{11} x_{1}+a_{12} x_{2}+\ldots \ldots+a_{1 n} x_{n}=b_{1} \\
a_{21} x_{1}+a_{22} x_{2}+\ldots \ldots+a_{2 n} x_{n}=b_{2} \\
\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\
a_{m 1} x_{1}+a_{m 2} x_{2}+\ldots \ldots+a_{m n} x_{n}=b_{m} .
\end{array}\)
Here X = A-1B,
where A = \(\left[\begin{array}{cccc}
a_{11} & a_{12} & \ldots \ldots \ldots & a_{1 n} \\
a_{21} & a_{22} & \ldots \ldots \ldots & a_{2 n} \\
\ldots & \ldots \ldots \ldots \ldots & \\
a_{m 1} & a_{m 2} \ldots \ldots \ldots . & a_{m n}
\end{array}\right]\), X = \(\left[\begin{array}{c}
x_{1} \\
x_{2} \\
\cdots \\
x_{n}
\end{array}\right]\), B = \(\left[\begin{array}{c}
b_{1} \\
b_{2} \\
\ldots \\
b_{m}
\end{array}\right]\)
(i) If |A| ≠ 0, then the system is consistent and has a unique solution.
(ii) If | A | = 0 and (adj. A) B = O, (O being a zero matrix) then the system is consistent and has infinitely many solutions.
(iii) If | A | = 0 and (adj. A) B ≠ O, then the system is inconsistent and has no solution.
9. SOLUTION OF HOMOGENEOUS EQUATIONS
To solve the equations :
a1x + b1y + c1z = 0
a2x + b2y + c1z = 0
a3x + b3y + c3z = 0.
Here AX = 0, where A = \(\left[\begin{array}{lll}
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2} \\
a_{3} & b_{3} & c_{3}
\end{array}\right]\) and X = \(\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]\)
(i) If |A| ≠ 0, then system has only trivial solution.
(ii) If |A| = 0, the system has infinitely many solutions.
Determinants Class 12 MCQs Questions with Answers
Question 1.
\(\left[\begin{array}{ccc}
1 & x & x^{2} \\
1 & y & y^{2} \\
1 & z & z^{2}
\end{array}\right]\)
(a) (x – y) (y + z)(z + x)
(b) (x + y) (y – z)(z – x)
(c) (x – y) (y – z)(z + x)
(d) (x – y) (y – z) (z – x)
Answer
Answer: (d) (x – y) (y – z) (z – x)
Question 2.
The value of the determinant
\(\left[\begin{array}{ccc}
3 & 1 & 7 \\
5 & 0 & 2 \\
2 & 5 & 3
\end{array}\right]\)
(a) 124
(b) 125
(c) 134
(d) 144
Answer
Answer: (c) 134
Question 3.
If a, b, c are in A.P. then the determinant
\(\left[\begin{array}{ccc}
x+2 & x+3 & x+2a \\
x+3 & x+4 & x+2b \\
x+4 & x+5 & x+2c
\end{array}\right]\)
(a) 1
(b) x
(c) 0
(d) 2x
Answer
Answer: (c) 0
Question 4.
If w is a non-real root of the equation x² – 1 = 0. then
\(\left[\begin{array}{ccc}
1 & ω & ω^{2} \\
ω & ω^{2} & 1 \\
ω^{2} & 1 & ω
\end{array}\right]\) =
(a) 0
(b) 1
(c) ω
(d) ω²
Answer
Answer: (a) 0
Question 5.
If Δ = \(\left[\begin{array}{cc}
10 & 2 \\
30 & 6
\end{array}\right]\) then A =
(a) 0
(b) 10
(c) 12
(d) 60
Answer
Answer: (a) 0
Question 6.
If 7 and 2 are two roots of the equation \(\left[\begin{array}{ccc}
x & 3 & 7 \\
2 & x & 2 \\
7 & 6 & x
\end{array}\right]\) then the third root is
(a) -9
(b) 14
(c) \(\frac{1}{2}\)
(d) None of these
Answer
Answer: (a) -9
Question 7.
If \(\left[\begin{array}{cc}
x & 2 \\
18 & x
\end{array}\right]\) = \(\left[\begin{array}{cc}
6 & 2 \\
18 & 6
\end{array}\right]\) x is equal to
(a) 6
(b) ±6
(c) -1
(d) -6
Answer
Answer: (b) ±6
Question 8.
\(\left[\begin{array}{ccc}
1 & a & a^{2}-bc \\
1 & b & b^{2}-ca \\
1 & c & c^{2}-ab
\end{array}\right]\) is equal to
(a) abc
(b) ab + bc + ca
(c) 0
(d) (a – b)(b – c)(c – a)
Answer
Answer: (c) 0
Question 9.
A = \(\left[\begin{array}{ll}
\alpha & q \\
q & \alpha
\end{array}\right]\) |A³| = 125 then α =
(a) ±3
(b) ±2
(c) ±5
(d) 0
Answer
Answer: (a) ±3
Question 10.
If a ≠ 0 and \(\left[\begin{array}{ccc}
1+a & 1 & 1 \\
1 & 1+a & 1 \\
1 & 1 & 1+a
\end{array}\right]\) = 0 then a =
(a) a = -3
(b) a = 0
(c) a = 1
(d) a = 3
Answer
Answer: (a) a = -3
Question 11.
If x > 0 and x ≠ 1. y > 0. and y ≠ 1, z > 0 and z ≠ 1 then
\(\left[\begin{array}{ccc}
1 & log_{x}y & log_{x}z \\
log_{y}x & 1 & log_{y}z \\
log_{z}x & log_{z}y & 1
\end{array}\right]\) is equal to
(a) 1
(b) -1
(c) 0
(d) None of these
Answer
Answer: (c) 0
Question 12.
\(\left[\begin{array}{ccc}
y+z & z & x \\
y & z+x & y \\
z & z & x+y
\end{array}\right]\) is equal to
(a) 6xyz
(b) xyz
(c) 4xyz
(d) xy + yz + zx
Answer
Answer: (c) 4xyz
Question 13.
If \(\left[\begin{array}{cc}
2 & 4 \\
5 & 1
\end{array}\right]\) = \(\left[\begin{array}{cc}
2x & 4 \\
6 & x
\end{array}\right]\) then the value of x is
(a) ±2
(b) ±\(\frac{1}{3}\)
(c) ±√3
(d) ± (0.5)
Answer
Answer: (c) ±√3
Question 14.
If \(\left[\begin{array}{cc}
2x & 5 \\
8 & x
\end{array}\right]\) = \(\left[\begin{array}{cc}
6 & -2 \\
7 & 3
\end{array}\right]\) then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6
Answer
Answer: (c) ±6
Question 15.
The value of determinant \(\left[\begin{array}{ccc}
a-b & b+c & a \\
b-c & c+a & b \\
c-a & a+b & c
\end{array}\right]\)
(a) a³ + b³ + c ³
(b) 3bc
(c) a³ + b³ + c³ – 3abc
(d) None of these
Answer
Answer: (c) a³ + b³ + c³ – 3abc
Question 16.
The area of a triangle with vertices (-3, 0) (3, 0) and (0, k) is 9 sq. units. The value of k will be
(a) 9
(b) 3
(c) -9
(d) 6
Answer
Answer: (b) 3
Question 17.
The determinant \(\left[\begin{array}{ccc}
b^{2}-ab & b-c & bc-ac \\
ab-a^{2} & a-b & b^{2}-ab \\
bc-ac & c-a & ab-a^{2}
\end{array}\right]\) equals
(a) abc(b – c)(c -a)(a – b)
(b) (b – c)(c – a)(a – b)
(c) (a + b + c)(b – c)(c – a)(a – b)
(d) None of these
Answer
Answer: (d) None of these
Question 18.
The number of distinct real roots of \(\left[\begin{array}{ccc}
sin x & cos x & cos x \\
cos x & sin x & cos x \\
cos x & cos x & sin x
\end{array}\right]\) = 0 in the interval –\(\frac{π}{4}\) ≤ x ≤ \(\frac{π}{4}\) is
(a) 0
(b) 2
(c) 1
(d) 3
Answer
Answer: (c) 1
Question 19.
If A, B and C are angles of a triangle, then the determinant
\(\left[\begin{array}{ccc}
-1 & cos C & cos B \\
cos C & -1 & cos A \\
cos B & cos A & -1
\end{array}\right]\)
(a) 0
(b) -1
(c) 1
(d) None of these
Answer
Answer: (a) 0
Question 20.
Let f(t) = \(\left[\begin{array}{ccc}
cot t & t & 1 \\
2 sin t & t & 2t \\
sin t & t & t
\end{array}\right]\) then \(_{t→0}^{lim}\) \(\frac{f(t)}{t^2}\) is equal to
(a) 0
(b) -1
(c) 2
(d) 3
Answer
Answer: (a) 0
Question 21.
The maximum value of \(\left[\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1+sin θ & 1 \\
1+cos θ & 1 & 1
\end{array}\right]\) is (θ is real number)
(a) \(\frac{1}{2}\)
(b) \(\frac{√3}{2}\)
(c) \(\frac{2√3}{4}\)
(d) √2
Answer
Answer: (a) \(\frac{1}{2}\)
Question 22.
If f(x) = \(\left[\begin{array}{ccc}
0 & x-a & x-b \\
x+a & 0 & x-c \\
x+b & x+c & 0
\end{array}\right]\) then
(a) f(a) = 0
(b) f(b) = 0
(c) f(0) = 0
(d) f(1) = 0
Answer
Answer: (c) f(0) = 0
Question 23.
If A = \(\left[\begin{array}{ccc}
2 & \lambda & -3 \\
0 & 2 & 5 \\
1 & 1 & 3
\end{array}\right]\) then A-1 exists if
(a) λ = 2
(b) λ ≠ 2
(c) λ ≠ -2
(d) None of these
Answer
Answer: (d) None of these
Question 24.
If A and B are invertible matrices, then which of the following is not correct?
(a) adj A = |A|.A-1
(b) det (a)-1 = [det (a)]-1
(c) (AB)-1 = B-1A-1
(d) (A + B)-1 = B-1 + A-1
Answer
Answer: (d) (A + B)-1 = B-1 + A-1
Question 25.
If x, y, z are all different from zero and
\(\left[\begin{array}{ccc}
1+x & 1 & 1 \\
1 & 1+y & 1 \\
1 & 1 & 1+z
\end{array}\right]\) = 0, then value of x-1 + y-1 + z-1 is
(a) xyz
(b) x-1y-1z-1
(c) -x – y – z
(d) -1
Answer
Answer: (d) -1
Question 26.
The value of the determinant \(\left[\begin{array}{ccc}
x & x+y & x+2y \\
x+2y & x & x+y \\
x+y & x+2y & x
\end{array}\right]\) is
(a) 9x² (x + y)
(b) 9y² (x + y)
(c) 3y² (x + y)
(d) 7x² (x + y)
Answer
Answer: (b) 9y² (x + y)
Question 27.
There are two values of a which makes determinant
Δ = \(\left[\begin{array}{ccc}
1 & -2 & 5 \\
2 & a & -1 \\
0 & 4 & 2a
\end{array}\right]\) = 86, then sum of these number is
(a) 4
(b) 5
(c) -4
(d) 9
Answer
Answer: (c) -4
Question 28.
Evaluate the determinant Δ = \(\left|\begin{array}{cc}
log_{3}512 & log_{4}3 \\
log_{3}8 & log_{4}9
\end{array}\right|\)
(a) \(\frac{15}{2}\)
(b) 12
(c) \(\frac{14}{3}\)
(d) 6
Answer
Answer: (a) \(\frac{15}{2}\)
Question 29.
\(\left|\begin{array}{cc}
x & -7 \\
x & 5 x+1
\end{array}\right|\)
(a) 3x² + 4
(b) x(5x + 8)
(c) 3x + 4x²
(d) x(3x + 4)
Answer
Answer: (b) x(5x + 8)
Question 30.
\( \left|\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \alpha
\end{array}\right|\)
(a) 0
(b) 1
(c) 2
(d) 3
Answer
Answer: (b) 1
Question 31.
\( \left|\begin{array}{ll}
\cos 15^{\circ} & \sin 15^{\circ} \\
\sin 75^{\circ} & \cos 75^{\circ}
\end{array}\right|\)
(a) 0
(b) 5
(c) 3
(d) 7
Answer
Answer: (a) 0
Question 32.
\(\left|\begin{array}{cc}
a+i b & c+i d \\
-c+i d & a-i b
\end{array}\right|\)
(a) (a + b)²
(b) (a + b + c + d)²
(c) (a² + b² – c² – d²)
(d) a² + b² + c² + a²
Answer
Answer: (d) a² + b² + c² + a²
Question 33.
If \(\left|\begin{array}{lll}
b+c & c+a & a+b \\
c+a & a+b & b+c \\
a+b & b+c & c+a
\end{array}\right|\) = \(k\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|\) then k =
(a) 0
(b) 1
(c) 2
(d) 3
Answer
Answer: (c) 2
Question 34.
If \(\left|\begin{array}{ccc}
a-b-c & 2 a & 2 a \\
2 b & b-c-a & 2 b \\
2 c & 2 c & c-a-b
\end{array}\right|\) = k (a + b + c)³ then k is
(a) 0
(b) 1
(c) 2
(d) 3
Answer
Answer: (b) 1
Question 35.
\(\left|\begin{array}{lll}
a+1 & a+2 & a+4 \\
a+3 & a+5 & a+8 \\
a+7 & a+10 & a+14
\end{array}\right|\) =
(a) 2
(b) -2
(c) 4
(d) -4
Answer
Answer: (b) -2
Question 36.
If abc ≠ 0 and \(\left|\begin{array}{ccc}
1+a & 1 & 1 \\
1 & 1+b & 1 \\
1 & 1 & 1+c
\end{array}\right|\) = 0 then \(\frac{1}{a}\) + \(\frac{1}{b}\) + \(\frac{1}{c}\) =
(a) 1
(b) 2
(c) -1
(d) -3
Answer
Answer: (c) -1
Question 37.
\(\left|\begin{array}{ccc}
2 x y & x^{2} & y^{2} \\
x^{2} & y^{2} & 2 x y \\
y^{2} & 2 x y & x^{2}
\end{array}\right|\) =
(a) (x³ + y³)²
(b) (x² + y²)³
(c) -(x² + y²)³
(d) -(x³ + y³)²
Answer
Answer: (d) -(x³ + y³)²
Question 38.
\(\left|\begin{array}{ccc}
b^{2} c^{2} & b c & b+c \\
c^{2} a^{2} & c a & c+a \\
a^{2} b^{2} & a b & a+b
\end{array}\right|\) =
(a) a7 + b7 + c7
(b) (a + b + c)7
(c) (a² + b² + c²) (a5 + b5 + c5)
(d) 0
Answer
Answer: (d) 0
Question 39.
If a, b, c are cube roots of unity, then
\(\left|\begin{array}{lll}
e^{a} & e^{2 a} & e^{3 a}-1 \\
e^{b} & e^{2 b} & e^{3 b}-1 \\
e^{c} & e^{2 c} & e^{3 c}-1
\end{array}\right|\) =
(a) 0
(b) e
(c) e²
(d) e³
Answer
Answer: (a) 0
Question 40.
The value of
\(\left|\begin{array}{ccc}
\cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\
\sin \alpha & \cos \alpha & \sin \beta \\
-\cos \alpha & \sin \alpha & \cos \beta
\end{array}\right|\)
is independent of
(a) α
(b) β
(c) α.β
(d) None of these
Answer
Answer: (a) α
Question 41.
If x is a complex root of the equation
\(\left|\begin{array}{lll}
1 & x & x \\
x & 1 & x \\
x & x & 1
\end{array}\right|\) + \(\left|\begin{array}{ccc}
1-x & 1 & 1 \\
1 & 1-x & 1 \\
1 & 1 & 1-x
\end{array}\right|\) = 0
then x2007 + x-2007 =
(a) 1
(b) -1
(c) -2
(d) 2
Answer
Answer: (c) -2
Question 42.
\(\left|\begin{array}{lll}
b-c & c-a & a-b \\
c-a & a-b & b-c \\
a-b & b-c & c-a
\end{array}\right|\) =
(a) 0
(b) 1
(c) 2
(d) 3
Answer
Answer: (a) 0
Question 43.
Let Δ = \(\left|\begin{array}{ccc}
x & y & z \\
x^{2} & y^{2} & z^{2} \\
x^{3} & y^{3} & z^{3}
\end{array}\right|\) then the value of Δ is
(a) (x – y) (y- z)(z – x)
(b) xyz
(c) x² + y² + z²)²
(d) xyz (x – y)(y – z) (z – x)
Answer
Answer: (d) xyz (x – y)(y – z) (z – x)
Question 44.
The value of the determinant \(\left|\begin{array}{ccc}
\alpha & \beta & \gamma \\
\alpha^{2} & \beta^{2} & \gamma^{2} \\
\beta+\gamma & \gamma+\alpha & \alpha+\beta
\end{array}\right|\)
(a) (α + β)(β + γ)(γ + α)
(b) (α – β)(β – γ) (γ – α) (α + β + γ)
(c) (α + β + γ)² (α – β – γ)²
(d) αβγ (α + β + γ)
Answer
Answer: (b) (α – β)(β – γ) (γ – α) (α + β + γ)