# CBSE Class 12 Maths –Chapter 4 Determinants- Study Materials

### 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.

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: (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

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

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

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

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

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

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)

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

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

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

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

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)

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

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: (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

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

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

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

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

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: (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

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

Question 24.
If A and B are invertible matrices, then which of the following is not correct?
(b) det (a)-1 = [det (a)]-1
(c) (AB)-1 = B-1A-1
(d) (A + B)-1 = B-1 + A-1

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

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: (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

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: (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)

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

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

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: (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

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

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

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

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

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

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

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

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

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

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: (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)