Solution of Chapter 4. Determinants (Mathemetics Part-I Book)

If then show that |2A| = 4|A|.

If then show that |3A| = 27|A|

Evaluate the determinants

If A = , find |A|.

Find values of x, if

If , then x is equal to

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

Which of the following is correct

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 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 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 sq units with vertices (2, –6), (5, 4) and (k, 4). Then k is

Write Minors and Cofactors of the elements of following determinants:

Write Minors and Cofactors of the elements of following determinants:

Write Minors and Cofactors of the elements of following determinants:

Write Minors and Cofactors of the elements of following determinants:

Using Cofactors of elements of second row, evaluate

Using Cofactors of elements of third column, evaluate.

If and A_ij is Cofactors of a_ij, then value of Δ is given by

Find adjoint of each of the matrices.

Verify A (adj A) = (adj A) A = |A|

Verify A (adj A) = (adj A) A = |A|

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

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

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

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

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

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

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

Let . Verify that (AB)^–1 = B^–1 A^–1.

If , show that A² – 5A + 7I = O. Hence find A^–1.

For the matrix , find the numbers a and b such that A² + aA + bI = O.

For the matrix

Show that A³– 6A² + 5A + 11 I = O. Hence, find A^–1.

If Verify that A³ – 6A² + 9A – 4I = O and hence find A^-1.

Let A be a non-singular 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

Write Minors and Cofactors of the elements of following determinants:

Write Minors and Cofactors of the elements of following determinants:

Write Minors and Cofactors of the elements of following determinants:

Write Minors and Cofactors of the elements of following determinants:

Using Cofactors of elements of second row, evaluate

Using Cofactors of elements of third column, evaluate.

If and A_ij is Cofactors of a_ij, then value of Δ is given by

Find adjoint of each of the matrices.

Verify A (adj A) = (adj A) A = |A|

Verify A (adj A) = (adj A) A = |A|

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

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

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

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

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

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

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

Let . Verify that (AB)^–1 = B^–1 A^–1.

If , show that A² – 5A + 7I = O. Hence find A^–1.

For the matrix , find the numbers a and b such that A² + aA + bI = O.

For the matrix

Show that A³– 6A² + 5A + 11 I = O. Hence, find A^–1.

If Verify that A³ – 6A² + 9A – 4I = O and hence find A^-1.

Let A be a non-singular 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 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.

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 +3 z = 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 , 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 is independent of θ.

Without expanding the determinant, prove that

Evaluate

If a, b and c are real numbers, and

Show that either a + b + c = 0 or a = b = c.

Solve the equation

Prove that

If , find (AB)^-1

Let . Verify that

[adj A]^-1 = adj (A^-1)

Let . Verify that

(A^-1)^-1 = A

Evaluate

Evaluate

Prove that

Prove that where p is any scalar.

Prove that

Prove that

Prove that

Solve the system of equations

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

If x, y, z are nonzero real numbers, then the inverse of matrix is

Let , where 0 ≤ θ ≤ 2π. Then