Chapter-4. Determinants

Choose the correct answer.

Let, where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]

Choose the correct answer.

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

A.  B. 

C.  D. 

Choose the correct answer.

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

A. 0 B. 1 C. x D. 2x

Solve the system of the following equations

Using properties of determinants, prove that:

Using properties of determinants, prove that:

Using properties of determinants, prove that:

Using properties of determinants, prove that:

Using properties of determinants, prove that:

Evaluate 

Evaluate 

:  

Let verify that

(i) 

(ii) 

If 

Prove that 

Solve the equations 

If a, b and c are real numbers, and,

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

Evaluate 

Without expanding the determinant, prove that

Prove that the determinant is independent of θ.

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.

If, find (A ^{- 1}). Using(A ^{- 1})

 solve the system of equations

Solve system of linear equations, using matrix method.

x - y + 2z = 7

3x + 4y - 5z = -5

2x - y + 3z = 12

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 + z = 4

2x + y - 3z = 0

x + y + z = 2

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

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.

Solve system of linear equations, using matrix method.

Examine the consistency of the system of equations.

5x - y + 4z = 5

2x + 3y + 5z = 2

5x - 2y + 6z = -1

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.

x +y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Examine the consistency of the system of equations.

2x - y = 5

x + y = 4

If A is an invertible matrix of order 2, then det(A ^{- 1})

 is equal toA. det (A) B.  C. 1 D. 0

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

A.  B.  C.  D. 

If verify that A³ - 6A² + 9A - 4I = O and hence find A ^{- 1}

For the matrixshow that A³ - 6A² + 5A + 11 I = O. Hence, find A ^{- 1.}

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

If, show that. Hence find.

Let and. Verify that 

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).

Verify A (adj A) = (adj A) A = I .

Verify A (adj A) = (adj A) A = I .

Find adjoint of each of the matrices.

Find adjoint of each of the matrices.

If  and Aij is Cofactors of aij, then value of Δ is given by

Using Cofactors of elements of third column, evaluate

(i)  (ii) 

Write Minors and Cofactors of the elements of following determinants:

(i)  (ii) 

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

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

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

Find values of k if area of triangle is 4 square units and vertices are

(i) (k, 0), (4, 0), (0, 2) (ii) (-2, 0), (0, 4), (0, k)

Show that points

are collinear

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

(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)

(iii) (-2, -3), (3, 2), (-1, -8)

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

Choose the correct answer.

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

A.  B.  C.  D. 

By using properties of determinants, show that:

By using properties of determinants, show that:

By using properties of determinants, show that:

By using properties of determinants, show that:

(i) 

(ii) 

By using properties of determinants, show that:

(i) 

(ii) 

By using properties of determinants, show that:

By using properties of determinants, show that:

(i) 

(ii) 

By using properties of determinants, show that:

By using properties of determinants, show that:

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

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

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

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

If, then x is equal to

(A) 6 (B) ±6 (C) - 6 (D) 0

Find values of x, if

(i)  (ii) 

Evaluate the determinants

(i)  (iii) 

(ii)  (iv) 

If, then show that

If, then show that

Evaluate the determinants in Exercises 1 and 2.

(i)  (ii) 

Evaluate the determinants in Exercises 1 and 2.