NCERT solutions for Class 12 Maths chapter 4 - Determinants [Latest edition]

Evaluate the determinants in Exercises 1 and 2. 

`|(2,4),(-5, -1)|`

Evaluate the determinants in Exercises 1 and 2. 

`|(cos theta, -sin theta),(sin theta, cos theta)|`

Evaluate the determinants in Exercises 1 and 2.

`|(x^2-x-1, x -1),(x+1, x+1)|`

if A = `[(1,2),(4,2)]` then show that |2A| = 4|A|

if A=`[(1,0,1),(0,1,2),(0,0,4)]` then show that `|3A| = 27|A|`

Evaluate the determinants `|(3,-1,-2),(0,0,-1),(3,-5,0)|`

Evaluate the determinants `|(0,1,2),(-1,0,-3),(-2,3,0)|`

Evaluate the determinants `|(3,-4,5),(1,1,-2),(2,3,1)|`

Evaluate the determinants

`|(2,-1,-2),(0,2,-1),(3,-5,0)|`

if A  = `[(1,1,-2),(2,1,-3),(5,4,-9)]`, Find |A|

Find values of x, if 2451=2x46x

Find values of x, if `|[2,3],[4,5]|=|[x,3],[2x,5]|`

if `|(x, 2),(18, x)| = |(6,2),(18,6)|`, then x is equal to 

(A) 6

(B) ±6

(C) −6

(D) 0

Using the property of determinants and without expanding, prove that:

`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`

Using the property of determinants and without expanding, prove that:

`|(a-b,b-c,c-a),(b-c,c-a,a-b),(a-a,a-b,b-c)| = 0`

Using the property of determinants and without expanding, prove that:

`|(2,7,65),(3,8,75),(5,9,86)| = 0`

Using the property of determinants and without expanding, prove that:

`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 0`

Using the property of determinants and without expanding, prove that:

`|(b+c, q+r, y+z),(c+a, r+p, z +x),(a+b, p+q, x + y )| = 2|(a,p,x),(b,q,y),(c, r,z)|`

By using properties of determinants, show that:

`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`

By using properties of determinants, show that:

`|(-a^2, ab, ac),(ba, -b^2, bc),(ca,cb, -c^2)| = 4a^2b^2c^2`

By using properties of determinants, show that:

`|(1,a,a^2),(1,b,b^2),(1,c,c^2)| = (a - b)(b-c)(c-a)`

By using properties of determinants, show that:

`|(1,1,1),(a,b,c),(a^3, b^3,c^3)|` = (a-b)(b-c)(c-a)(a+b+c)

By using properties of determinants, show that:

`|(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)| = (x-y)(y-z)(z-x)(xy+yz+zx)`

By using properties of determinants, show that:

`|(x+4,2x,2x),(2x,x+4,2x),(2x , 2x, x+4)| = (5x + 4)(4-x)^2`

By using properties of determinants, show that:

`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`

By using properties of determinants, show that:

`|(a-b-c, 2a,2a),(2b, b-c-a,2b),(2c,2c, c-a-b)| = (a + b + c)^2`

By using properties of determinants, show that:

`|(x+y+2z, x, y),(z, y+z+2z,y),(z,x,z+x+2y)| = 2(x+y+z)^3`

By using properties of determinants, show that:

`|(1,x,x^2),(x^2,1,x),(x,x^2,1)| = (1-x^3)^2`

By using properties of determinants, show that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1+a^2+b^2)`

By using properties of determinants, show that:

`|(a^2+1, ab, ac),(ab, b^2+1, bc),(ca, cb, c^2+1)| = 1+a^2+b^2+c^2`

Let A be a square matrix of order 3 × 3, then | kA| is equal to

(A) k|A|

(B) k² | A |

(C) k³ | A |

(D) 3k | A |

Which of the following is correct?

A. Determinant is a square matrix.

B. Determinant is a number associated to a matrix.

C. Determinant is a number associated to a square matrix.

D. None of these

Find area of the triangle with vertices at the point given in each of the following:

(1, 0), (6, 0), (4, 3)

Find area of the triangle with vertices at the point given in each of the following:

(2, 7), (1, 1), (10, 8)

Find area of the triangle with vertices at the point given in each of the following:

(−2, −3), (3, 2), (−1, −8)

Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Find values of k if area of triangle is 4 square units and vertices are (k, 0), (4, 0), (0, 2)

Find values of k if area of triangle is 4 square units and vertices are (−2, 0), (0, 4), (0, k)

Find equation of line joining (1, 2) and (3, 6) using determinants

Find equation of line joining (3, 1) and (9, 3) using determinants

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is

A. 12

B. −2

C. −12, −2

D. 12, −2

Write Minors and Cofactors of the elements of following determinants:

`|(2,-4),(0,3)|`

Write Minors and Cofactors of the elements of following determinants:

`|(a,c),(b,d)|`

Write Minors and Cofactors of the elements of following determinants:

`|(1,0,0),(0,1,0),(0,0,1)|`

Write Minors and Cofactors of the elements of following determinants:

`|(1,0,4),(3,5,-1),(0,1,2)|`

Using Cofactors of elements of second row, evaluate `triangle = |(5,3,8),(2,0,1),(1,2, 3)|`

Using Cofactors of elements of third column, evaluate `triangle = |(1,x,yz),(1,y,zx),(1,z,xy)|`

if `triangle = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` and A_ij is Cofactors of a_ij, then value of Δ is given by

(A) a₁₁ A₃₁+ a₁₂ A₃₂ + a₁₃ A₃₃

(B) a₁₁ A₁₁+ a₁₂ A₂₁ + a₁₃ A₃₁

(C) a₂₁ A₁₁+ a₂₂ A₁₂ + a₂₃ A₁₃

(D) a₁₁ A₁₁+ a₂₁ A₂₁ + a₃₁ A₃₁

Find adjoint of each of the matrices.'

`[(1,2),(3,4)]`

Find adjoint of each of the matrices.

`[(1,-1,2),(2,3,5),(-2,0,1)]`

Verify A (adj A) = (adj A) A = |A|I

`[(2,3),(-4,-6)]`

Verify A (adj A) = (adj A) A = |A|I

`[(1,-1,2),(3,0,-2),(1,0,3)]`

Find the inverse of each of the matrices (if it exists).

`[(2,-2),(4,3)]`

Find the inverse of each of the matrices (if it exists).

`[(-1,5),(-3,2)]`

Find the inverse of each of the matrices (if it exists).

`[(1,2,3),(0,2,4),(0,0,5)]`

Find the inverse of each of the matrices (if it exists).

`[(1,0,0),(3,3,0),(5,2,-1)]`

Find the inverse of each of the matrices (if it exists).

`[(2,1,3),(4,-1,0),(-7,2,1)]`

Find the inverse of each of the matrices (if it exists).

`[(1,-1,2),(0,2,-3),(3,-2,4)]`

Find the inverse of each of the matrices (if it exists).

`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`

Let `A =[(3,7),(2,5)] and B = [(6,8),(7,9)]`. Verify that `(AB)^(-1) = B^(-1)A^(-1)`

if A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = O. Hence find A^–1.

For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A² + aA + bI = O.

For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A³ − 6A² + 5A + 11 I = O. Hence, find A^−1.

if A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A³ − 6A² + 9A − 4I = O and hence find A⁻¹

Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to

(A) |A |

(B) | A|²

(C) | A|³

(D) 3|A|

If A is an invertible matrix of order 2, then det (A⁻¹) is equal to

(A) det (A)

(B) 1/det (A)

(C) 1

(D) 0

Examine the consistency of the system of equations.

x + 2y = 2

2x + 3y = 3

Examine the consistency of the system of equations.

2x − y = 5

x + y = 4

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Examine the consistency of the system of equations.

x + y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

Solve system of linear equations, using matrix method.

5x + 2y = 4

7x + 3y = 5

Solve system of linear equations, using matrix method.

2x – y = –2

3x + 4y = 3

Solve system of linear equations, using matrix method.

4x – 3y = 3

3x – 5y = 7

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Solve system of linear equations, using matrix method.

2x + y + z = 1

x – 2y – z = 3/2

3y – 5z = 9

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Solve system of linear equations, using matrix method.

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A⁻¹. Using A⁻¹ solve the system of equations

2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.

Without expanding the determinant, prove that

`|(a, a^2,bc),(b,b^2, ca),(c, c^2,ab)| = |(1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3)|`

Evaluate `|(cos alpha cos beta, cos alpha sin beta, -sin alpha),(-sin beta, cos beta, 0),(sin alpha cos beta, sin alpha sin beta,cos alpha )|`

If a, b and c are real numbers, and triangle =`|(b+c, c+a, a+b),(c+a,a+b, b+c),(a+b, b+c, c+a)|` = 0 Show that either a + b + c = 0 or a = b = c.

Solve the equations `|(x+a,x,x),(a,x+a,x),(x,x,x+a)| = 0, a != 0`

Prove that `|(a^2, bc, ac+c^2),(a^2+ab, b^2, ac),(ab, b^2+bc, c^2)| = 4a^2b^2c^2`

if A^(-1) =`[(3,-1,1),(-15,6,-5),(5,-2,2)] and B = [(1,2,-2),(-1,3,0),(0,-2,1)] " find " (AB)^(-1)`

Let A = `[(1,-2,1),(-2,3,1),(1,1,5)]` verify that

1) [adj A]^–1 = adj (A^–1)

2) (A^–1)^–1 = A

Evaluate `|(x, y, x+y),(y, x+y, x),(x+y, x, y)|`

Evaluate `|(1,x,y),(1,x+y,y),(1,x,x+y)|`

Using properties of determinants, prove that:

`|(alpha, alpha^2,beta+gamma),(beta, beta^2, gamma+alpha),(gamma, gamma^2, alpha+beta)|` =  (β – γ) (γ – α) (α – β) (α + β + γ)

Using properties of determinants, prove that:

`|(x, x^2, 1+px^3),(y, y^2, 1+py^3),(z, z^2, 1+pz^2)|` = (1 + pxyz) (x – y) (y – z) (z – x), where p is any scalar.

Using properties of determinants, prove that:

`|(3a, -a+b, -a+c),(-b+a, 3b, -b+c),(-c+a, -c+b, 3c)|`= 3(a + b + c) (ab + bc + ca)

Using properties of determinants, prove that:

`|(1, 1+p, 1+p+q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| =  1`                 

Using properties of determinants, prove that

`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`

Solve the system of the following equations

`2/x+3/y+10/z = 4`

`4/x-6/y + 5/z = 1`

`6/x + 9/y - 20/x = 2`

Choose the correct answer.

If a, b, c, are in A.P., then the determinant

`|(x+2, x+3,x +2a),(x+3,x+4,x+2b),(x+4,x+5,x+2c)|`

A. 0

B. 1

C. x

D. 2x

Choose the correct answer.

If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is

A) `[(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))]`

B) `xyz[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`

c) `1/xyz[(x,0,0),(0,y,0),(0,0,z)]`

D) `1/xyz [(1,0,0),(0,1,0),(0,0,1)]`

Choose the correct answer.

Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]