Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:

Evaluate the following determinants:

i.

ii.

iii.

iv.

Evaluate

Show that

Evaluate by two methods.

Evaluate :

Evaluate :

If and , verify that |AB| = |A| |B|.

If , then show that .

Find the value of x, if

Find the value of x, if

Find the value of x, if

Find the value of x, if

Find the value of x, if

Find the value of x, if

Find the integral value of x, if

For what value of x matrix A is singular?

For what value of x matrix A is singular?

Evaluate the following determinant:

Evaluate the following determinant:

Evaluate the following determinant:

Evaluate the following determinant:

Evaluate the following determinant:

Evaluate the following determinant:

Evaluate the following determinant:

Evaluate the following determinant:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

Without expanding, show that the value of each of the following determinants is zero:

, where A, B, C are the angles of ΔABC.

Evaluate the following:

Evaluate the following:

Evaluate the following:

Evaluate the following:

Evaluate the following:

Evaluate the following:

Evaluate the following:

If ,, then prove that Δ + Δ₁ = 0.

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

=

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities:

Prove the following identities –

Prove the following identities –

= (a + b - c)(b + c - a)(c + a - b)

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

= 2(a + b)(b + c)(c + a)

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

= abc(a² + b² + c²)³

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

Prove the following identities –

= 2(a - b)(b - c)(c - a)(a + b + c)

Without expanding, prove that .

Show that where a,b,c are in A.P.

Show that where α, β and γ are in A.P.

If a, b, c are real numbers such that , then show that either a + b + c = 0 or a = b = c.

If , find the value of .

Show that x = 2 is a root of the equation and solve it completely.

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

Solve the following determinant equations:

If a, b and c are all non-zero and , then prove that .

If , then using properties of determinants, find the value of , where .

Find the area of the triangle with vertices at the points:

(3, 8), (– 4, 2) and (5, – 1)

Find the area of the triangle with vertices at the points:

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

Find the area of the triangle with vertices at the points:

(– 1, – 8), (– 2, – 3) and (3, 2)

Find the area of the triangle with vertices at the points:

(0, 0) (6, 0) and (4, 3)

Using determinants show that the following points are collinear:

(5, 5), (– 5, 1) and (10, 7)

Using determinants show that the following points are collinear:

(1, – 1), (2, 1) and (4, 5)

Using determinants show that the following points are collinear:

(3, – 2), (8, 8) and (5, 2)

Using determinants show that the following points are collinear:

(2, 3), (– 1, – 2) and (5, 8)

If the points (a, 0), (o, b) and (1, 1) are collinear, prove that a + b = ab.

Using determinants prove that the points (a, b) (a’, b’) and (a – a’, b – b’) are collinear if ab’ = a’b.

Find the value of λ so that the points (1, – 5), (– 4, 5) and (λ, 7) are collinear.

Find the value of x if the area of a triangle is 35 square cms with vertices (x, 4), (2, – 6) and (5, 4).

Using determinants, find the area of a triangle whose vertices are (1, 4), (2, 3) and (– 5, – 3). Are the given points collinear?

Using determinants, find the area of the triangle with vertices (– 3, 5), (3, – 6) and (7, 2).

Using determinants, find the value of k so that the points (k, 2 – 2 k), (– k + 1, 2k) and (– 4 – k, 6 – 2k) may be collinear.

If the points (x, 2), (5, – 2) and (8, 8) are collinear, find x using determinants.

If the points (3, – 2), (x,2) and (8,8) are collinear, find x using determinant.

Using determinants, find the equation of the line joining the points

(1, 2) and (3, 6)

Using determinants, find the equation of the line joining the points

(3, 1) and (9, 3)

Find values of K, if the area of a triangle is 4 square units whose vertices are

(k,0), (4,0) and (0,2)

Find values of K, if the area of a triangle is 4 square units whose vertices are

(– 2,0), (0, 4) and (0, k)

Solve the following systems of linear equations by Cramer’s rule:

x – 2y = 4

– 3x + 5y = – 7

Solve the following systems of linear equations by Cramer’s rule:

2x – y = 1

7x – 2y = – 7

Solve the following systems of linear equations by Cramer’s rule:

2x – y = 17

3x + 5y = 6

Solve the following systems of linear equations by Cramer’s rule:

3x + y = 19

3x – y = 23

Solve the following systems of linear equations by Cramer’s rule:

2x – y = – 2

3x + 4y = 3

Solve the following systems of linear equations by Cramer’s rule:

3x + ay = 4

2x + ay = 2, a≠0

Solve the following systems of linear equations by Cramer’s rule:

2x + 3y = 10

x + 6y = 4

Solve the following systems of linear equations by Cramer’s rule:

5x + 7y = – 2

4x + 6y = – 3

Solve the following systems of linear equations by Cramer’s rule:

9x + 5y = 10

3y – 2x = 8

Solve the following systems of linear equations by Cramer’s rule:

x + 2y = 1

3x + y = 4

Solve the following system of the linear equations by Cramer’s rule:

3x + y + z = 2

2x – 4y + 3z = – 1

4x + y – 3z = – 11

Solve the following system of the linear equations by Cramer’s rule:

x – 4y – z = 11

2x – 5y + 2z = 39

– 3x + 2y + z = 1

Solve the following system of the linear equations by Cramer’s rule:

6x + y – 3z = 5

X + 3y – 2z = 5

2x + y + 4z = 8

Solve the following system of the linear equations by Cramer’s rule:

x + y = 5

y + z = 3

x + z = 4

Solve the following system of the linear equations by Cramer’s rule:

2y – 3z = 0

X + 3y = – 4

3x + 4y = 3

Solve the following system of the linear equations by Cramer’s rule:

5x – 7y + z = 11

6x – 8y – z = 15

3x + 2y – 6z = 7

Solve the following system of the linear equations by Cramer’s rule:

2x – 3y – 4z = 29

– 2x + 5y – z = – 15

3x – y + 5z = – 11

Solve the following system of the linear equations by Cramer’s rule:

x + y = 1

x + z = – 6

x – y – 2z = 3

Solve the following system of the linear equations by Cramer’s rule:

x + y + z + 1 = 0

ax + by + cz + d = 0

a²x + b²y + c²z + d² = 0

Solve the following system of the linear equations by Cramer’s rule:

x + y + z + w = 2

x – 2y + 2z + 2w = – 6

2x + y – 2z + 2w = – 5

3x – y + 3z – 3w = – 3

Solve the following system of the linear equations by Cramer’s rule:

2x – 3z + w = 1

x – y + 2w = 1

– 3y + z + w = 1

x + y + z = 1

Show that each of the following systems of linear equations is inconsistent:

2x – y = 5

4x – 2y = 7

Show that each of the following systems of linear equations is inconsistent:

3x + y = 5

– 6x – 2y = 9

Show that each of the following systems of linear equations is inconsistent:

3x – y + 2z = 3

2x + y + 3z = 5

x – 2y – z = 1

Show that each of the following systems of linear equations is inconsistent:

x + y + z = 3

2x – y + z = 2

3x + 6y + 5z = 20.

Show that each of the following systems of linear equations has infinite number of solutions and solve:

x – y + z = 3

2x + y – z = 2

– x – 2y + 2z = 1

Show that each of the following systems of linear equations has infinite number of solutions and solve:

x + 2y = 5

3x + 6y = 15

Show that each of the following systems of linear equations has infinite number of solutions and solve:

x + y – z = 0

x – 2y + z = 0

3x + 6y – 5z = 0

Show that each of the following systems of linear equations has infinite number of solutions and solve:

2x + y – 2z = 4

x – 2y + z = – 2

5x – 5y + z = – 2

Show that each of the following systems of linear equations has infinite number of solutions and solve:

x – y + 3z = 6

x + 3y – 3z = – 4

5x + 3y + 3z = 10

A salesman has the following record of sales during three months for three items A,B and C which have different rates of commission.

Find out the rates of commission on items A,B and C by using determinant method.

An automobile company uses three types of steel S₁, S₂ and S₃ for producing three types of cars C₁, C₂ and C₃. Steel requirements (in tons) for each type of cars are given below:

Using Cramer’s rule, find the number of cars of each type which can be produced using 29, 13 and 16 tonnes of steel of three types respectively.

Solve each of the following systems of homogeneous linear equations:

x + y – 2z = 0

2x + y – 3z = 0

5x + 4y – 9z = 0

Solve each of the following systems of homogeneous linear equations:

2x + 3y + 4z = 0

X + y + z = 0

2x + 5y – 2z = 0

Solve each of the following systems of homogeneous linear equations:

3x + y + z = 0

x – 4y3z = 0

2x + 5y – 2z = 0

Find the real values of λ for which the followings system of linear equations has non – trivial solutions. Also, find the non – trivial solutions

2λx – 2y + 3z = 0

X + λy + 2z = 0

2x + λz = 0

If a,b,c are non – zero real numbers and if the system of equations

(a – 1) x = y + z

(b – 1) y = z + x

(c – 1) z = x + y

Has a non – trivial solution, then prove that ab + bc + ca = abc.

Mark the correct alternative in the following:

If A and B are square matrices or order 2, then det (A + B) = 0 is possible only when

Mark the correct alternative in the following:

Which of the following is not correct?

Mark the correct alternative in the following:

If and C_ij is cofactor of a_ij in A, then value of |A| is given by

Mark the correct alternative in the following:

Which of the following is not correct in a given determinant of A, where A = [a_ij]_{3 × 3}.

Mark the correct alternative in the following:

Let . Then, the value of 5a + 4b + 3c + 2d + e is equal to

Mark the correct alternative in the following:

The value of the determinant is independent of

Mark the correct alternative in the following:

If , then

Mark the correct alternative in the following:

If and , then n equals

Mark the correct alternative in the following:

Let be an identify in x, were a, b, c, d, e are independent of x. Then the value of e is

Mark the correct alternative in the following:

Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant . The other factor in the value of the determinant is

Mark the correct alternative in the following:

If a, b, c are distinct then the value of x satisfying is

Mark the correct alternative in the following:

If the determinant , then

Mark the correct alternative in the following:

If ω is a non-real cube root of unity and n is not a multiple of 3, then is equal to

Mark the correct alternative in the following:

If , then the value of is

Mark the correct alternative in the following:

If a > 0 and discriminant of ax² + 2bx + c is negative, then is

Mark the correct alternative in the following:

The value of is

Mark the correct alternative in the following:

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

If A + B + C = π, then the value of is equal to

Mark the correct alternative in the following:

The number of distinct real roots of lies in the interval is

Mark the correct alternative in the following:

Let , where 0 ≤ θ ≤ 2π. Then,

Mark the correct alternative in the following:

If , then x =

Mark the correct alternative in the following:

If , then

Mark the correct alternative in the following:

The value of the determinant is

Mark the correct alternative in the following:

If x, y, z are different from zero and , then the value of is

Mark the correct alternative in the following:

The determinant equals

Mark the correct alternative in the following:

If x, y ∈ R, then the determinant lies in the interval

Mark the correct alternative in the following:

The maximum value of is (θ is real)

Mark the correct alternative in the following:

The value of the determinant is

Mark the correct alternative in the following:

Let , then is equal to

Mark the correct alternative in the following:

There are two values of a which makes the determinant equal to 86. The sum of these two values is

Mark the correct alternative in the following:

If , t hen the value of is

Mark the correct alternative in the following:

The value of is

If A is a singular matrix, then write the value of |A|.

For what value of x, the matrix is singular?

State whether the matrix is singular or non-singular.

Find the value of the determinant .

Find the value of the determinant .

Write the value of the determinant .

If and , find the value of |A| + |B|.

If and , find |AB|.

Evaluate: .

If w is an imaginary cube root of unity, find the value of .

If and , find .

If A = [a_ij] is a 3 × 3 diagonal matrix such that a₁₁ = 1, a₂₂ = 2 and a₃₃ = 3, then find |A|.

If A = [a_ij] is a 3 × 3 scalar matrix such that a₁₁ = 2, then write the value of |A|.

If I₃ denotes identity matrix of order 3 × 3, write the value of its determinant.

A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?

On expanding by first row, the value of the determinant of 3 × 3 square matrix A = [a_ij] is a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃, where C_ij is the cofactor of a_ij is the cofactor of a_ij in A. Write the expression for its value of expanding by second column.

Let A = [a_ij] be a square matrix of order 3 × 3 and C_ij denote cofactor of a_ij in A. If |A| = 5, write the value of a₃₁C₃₁ + a₃₂C₃₂ + a₃₃C₃₃.

In question 18, write the value of a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃.

Write the value of .

If A is a square matrix satisfying A^TA = I, write the value of |A|.

If A and B are square matrices of the same order such that |A| = 3 and AB = I, then write the value of |B|.

A is skew-symmetric of order 3, write the value of |A|.

If A is a square matrix of order 3 with determinant 4, then write the value of |-A|.

If A is a square matrix such that |A| = 2, write the value of |AA^T|.

Find the value of the determinant .

Write the value of the determinant .

If the matrix is singular, find the value of x.

If A is a square matrix of order n × n such that |A| = λ, then write the value of |-A|.

Find the value of the determinant .

If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.

A matrix of order 3 × 3 has determinant 2. What is the value of |A(3I)|, where I is the identity matrix of order 3 × 3.

If A and B are square matrices of order 3 such that |A| = -1, |B| = 3, then find the value of |3AB|.

Write the value of .

Write the cofactor of a₁₂ in the matrix .

If find x.

Find the value of x from the following: .

Write the value of the determinant .

If |A| = 2, where A is 2 × 2 matrix, find |adj A|.

For what is the value of the determinant ?

For what value of x is the matrix singular?

A matrix A of order 3 × 3 is such that |A| = 4. Find the value of |2A|.

Evaluate:

If . Write the cofactor of the element a₃₂.

If , then write the value of x.

If , then write the value of x.

If , find the value of x.

If , write the value of x.

If A is a 3 × 3 matrix, |A| ≠ 0 and |3A| = k|A| then write value of k.

Write the value of the determinant .

Write the value of the determinant .

If , then for any natural number, find the value of Det(Aⁿ).

Find the maximum value of .

If x ∈ N and , then find the value of x.

If , write the value of x.

If A is a 3 × 3 matrix, then what will be the value of k if Det(A^-1) = (Det A)^k?