NCERT solutions for Mathematics Part 1 and 2 [English] Class 12 chapter 4 - Determinants [Latest edition]

Evaluate the following determinant.

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

Evaluate the following determinant.

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

Evaluate the following determinant.

`|(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 determinant.

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

Evaluate the determinant.

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

Evaluate the determinant.

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

Evaluate the determinant.

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

If A = `[(1,1,-2),(2,1,-3),(5,4,-9)]`, find |A|.

Find the value of x, if `|(2,4),(5,1)|=|(2x, 4), (6,x)|`.

Find the value 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 ______.

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

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

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

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

Find the area of a triangle with vertices at the point given in 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 sq. units and vertices are (k, 0), (4, 0), (0, 2).

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

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

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

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

Write Minors and Cofactors of the elements of the following determinant:

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

Write Minors and Cofactors of the elements of the following determinant:

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

Write Minors and Cofactors of the elements of the following determinant:

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

Write Minors and Cofactors of the elements of the following determinant:

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

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

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

If Δ = `|(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 the value of Δ is given by ______.

Find the adjoint of the matrices.

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

Find the adjoint 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 the matrices (if it exists).

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

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

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

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

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

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

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

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

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

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

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

Find the inverse 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⁻¹A⁻¹.

If A = `[(3,1),(-1,2)]` show that A² – 5A + 7I = 0. Hence, find A^–1.

For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A² + aA + bI = 0.

For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A³ − 6A² + 5A + 11 I = 0. Hence, find A⁻¹.

If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A³ − 6A² + 9A − 4I = 0 and hence find A⁻¹.

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

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

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 the system of linear equations using the matrix method.

5x + 2y = 4

7x + 3y = 5

Solve the system of linear equations using the matrix method.

2x – y = –2

3x + 4y = 3

Solve the system of linear equations using the matrix method.

4x – 3y = 3

3x – 5y = 7

Solve the system of linear equations using the matrix method.

5x + 2y = 3

3x + 2y = 5

Solve the system of linear equations using the matrix method.

2x + y + z = 1

x – 2y – z = `3/2`

3y – 5z = 9

Solve the system of linear equations using the matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Solve the system of linear equations using the matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Solve the system of linear equations using the 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 the 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

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

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