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Solutions

Overview

Step-by-step NCERT solutions for Determinants (Chapter 4, CBSE Class 12 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Determinants textbook chapter.

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What these solutions cover

All 61 questions in Determinants are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 4.1

  1. Evaluate the determinant: |2 4| |-5 -1|
  2. Evaluate the determinants:
    • (i) |cos θ −sin θ| |sin θ cos θ|
    • (ii) |x² − x + 1 x − 1| |x + 1 x + 1|
  3. If A = [[1, 2], [4, 2]], then show that |2A| = 4|A|.
  4. If A = [[1, 0, 1], [0, 1, 2], [0, 0, 4]], then show that |3A| = 27|A|.
  5. Evaluate the determinants:
    • (i) |3 −1 −2| |0 0 −1| |3 −5 0|
    • (ii) |3 −4 5| |1 1 −2| |2 3 1|
    • (iii) |0 1 2| |−1 0 −3| |−2 3 0|
    • (iv) |2 −1 −2| |0 2 −1| |3 −5 0|
  6. If A = [[1, 1, −2], [2, 1, −3], [5, 4, −9]], find |A|.
  7. Find values of x, if:
    • (i) |2 4| = |2x 4| |5 1| |6 x|
    • (ii) |2 3| = |x 3| |4 5| |2x 5|
  8. If |x 2 | = |6 2|, then x is equal to |18 x| |18 6| (A) 6 (B) ±6 (C) −6 (D) 0

Exercise 4.2

  1. 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)
  2. Show that points A(a, b+c), B(b, c+a), C(c, a+b) are collinear.
  3. Find values of k if area of triangle is 4 sq. units and vertices are:
    • (i) (k, 0), (4, 0), (0, 2)
    • (ii) (−2, 0), (0, 4), (0, k)
  4. 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.
  5. If area of triangle is 35 sq units with vertices (2, −6), (5, 4) and (k, 4). Then k is (A) 12 (B) −2 (C) −12, −2 (D) 12, −2

Exercise 4.3

  1. Write Minors and Cofactors of the elements of following determinants:
    • (i) |2 −4| |0 3|
    • (ii) |a c| |b d|
  2. Write Minors and Cofactors of the elements of following determinants:
    • (i) |1 0 0| |0 1 0| |0 0 1|
    • (ii) |1 0 4| |3 5 −1| |0 1 2|
  3. Using Cofactors of elements of second row, evaluate ∆ = |5 3 8| |2 0 1| |1 2 3|
  4. Using Cofactors of elements of third column, evaluate ∆ = |1 x yz| |1 y zx| |1 z xy|
  5. If ∆ = |a11 a12 a13| |a21 a22 a23| and A_ij is Cofactors of a_ij, then value of ∆ is given by |a31 a32 a33| (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

Exercise 4.4

  1. Find adjoint of each of the matrices: [[1, 2], [3, 4]]
  2. Find adjoint of each of the matrices: [[1, −1, 2], [2, 3, 5], [−2, 0, 1]]
  3. Verify A(adj A) = (adj A)A = |A|I for A = [[2, 3], [−4, −6]].
  4. Verify A(adj A) = (adj A)A = |A|I for A = [[1, −1, 2], [3, 0, −2], [1, 0, 3]].
  5. Find the inverse of the matrix (if it exists): [[2, −2], [4, 3]]
  6. Find the inverse of the matrix (if it exists): [[−1, 5], [−3, 2]]
  7. Find the inverse of the matrix (if it exists): [[1, 2, 3], [0, 2, 4], [0, 0, 5]]
  8. Find the inverse of the matrix (if it exists): [[1, 0, 0], [3, 3, 0], [5, 2, −1]]
  9. Find the inverse of the matrix (if it exists): [[2, 1, 3], [4, −1, 0], [−7, 2, 1]]
  10. Find the inverse of the matrix (if it exists): [[1, −1, 2], [0, 2, −3], [3, −2, 4]]
  11. Find the inverse of the matrix (if it exists): [[1, 0, 0 ], [0, cos α, sin α ], [0, sin α, −cos α]]
  12. Let A = [[3, 7], [2, 5]] and B = [[6, 8], [7, 9]]. Verify that (AB)⁻1 = B⁻1 A⁻1.
  13. If A = [[3, 1], [−1, 2]], show that A² − 5A + 7I = O. Hence find A⁻1.
  14. For the matrix A = [[3, 2], [1, 1]], find the numbers a and b such that A² + aA + bI = O.
  15. For the matrix A = [[1, 1, 1], [1, 2, −3], [2, −1, 3]], show that A³ − 6A² + 5A + 11I = O. Hence find A⁻1.
  16. If A = [[2, −1, 1], [−1, 2, −1], [1, −1, 2]], verify that A³ − 6A² + 9A − 4I = O and hence find A⁻1.
  17. 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|
  18. 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

Exercise 4.5

  1. Examine the consistency of the system of equations: x + 2y = 2 2x + 3y = 3
  2. Examine the consistency of the system of equations: 2x − y = 5 x + y = 4
  3. Examine the consistency of the system of equations: x + 3y = 5 2x + 6y = 8
  4. Examine the consistency of the system of equations: x + y + z = 1 2x + 3y + 2z = 2 ax + ay + 2az = 4
  5. Examine the consistency of the system of equations: 3x − y − 2z = 2 2y − z = −1 3x − 5y = 3
  6. Examine the consistency of the system of equations: 5x − y + 4z = 5 2x + 3y + 5z = 2 5x − 2y + 6z = −1
  7. Solve system of linear equations using matrix method: 5x + 2y = 4 7x + 3y = 5
  8. Solve system of linear equations using matrix method: 2x − y = −2 3x + 4y = 3
  9. Solve system of linear equations using matrix method: 4x − 3y = 3 3x − 5y = 7
  10. Solve system of linear equations using matrix method: 5x + 2y = 3 3x + 2y = 5
  11. Solve system of linear equations using matrix method: 2x + y + z = 1 x − 2y − z = 3/2 3y − 5z = 9
  12. Solve system of linear equations using matrix method: x − y + z = 4 2x + y − 3z = 0 x + y + z = 2
  13. Solve system of linear equations using matrix method: 2x + 3y + 3z = 5 x − 2y + z = −4 3x − y − 2z = 3
  14. Solve system of linear equations using matrix method: x − y + 2z = 7 3x + 4y − 5z = −5 2x − y + 3z = 12
  15. 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
  16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ₹60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is ₹90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is ₹70. Find cost of each item per kg by matrix method.

Miscellaneous Exercise

  1. Prove that the determinant |x sin θ cos θ| |−sin θ −x 1 | |cos θ 1 x | is independent of θ.
  2. Evaluate: |cos α cos β cos α sin β −sin α| |−sin β cos β 0 | |sin α cos β sin α sin β cos α|
  3. 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.
  4. Let A = [[1, 2, 1], [2, 3, 1], [1, 1, 5]]. Verify that:
    • (i) [adj A]⁻1 = adj(A⁻1)
    • (ii) (A⁻1)⁻1 = A
  5. Evaluate: |x y x+y| |y x+y x | |x+y x y |
  6. Evaluate: |1 x y | |1 x+y y | |1 x x+y|
  7. Solve the system of equations: 2/x + 3/y + 10/z = 4 4/x − 6/y + 5/z = 1 6/x + 9/y − 20/z = 2
  8. 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]]
  9. Let A = [[1, sin θ, 1], [−sin θ, 1, sin θ], [−1, −sin θ, 1]], where 0 ≤ θ ≤ 2π. Then (A) Det(A) = 0 (B) Det(A) ∈ (2, ∞) (C) Det(A) ∈ (2, 4) (D) Det(A) ∈ [2, 4]
Sample

Worked examples

3 questions from Determinants, solved step by step.

Exercise 4.1

Evaluate the determinants: (i) |cos θ −sin θ| |sin θ cos θ| (ii) |x² − x + 1 x − 1| |x + 1 x + 1|

(i) det = cos θ · cos θ − (−sin θ) · sin θ = cos²θ + sin²θ = 1.

(ii) det = (x² − x + 1)(x + 1) − (x − 1)(x + 1).

Expand: (x² − x + 1)(x + 1) = x³ + x² − x² − x + x + 1 = x³ + 1.

(x − 1)(x + 1) = x² − 1.

det = x³ + 1 − (x² − 1) = x³ − x² + 2.

Answer: (i) 1 (ii) x³ − x² + 2

Exercise 4.3

Write Minors and Cofactors of the elements of following determinants: (i) |2 −4| |0 3| (ii) |a c| |b d|

(i) For the determinant with elements a11=2, a12=−4, a21=0, a22=3:

Minors: M11 = 3, M12 = 0, M21 = −4, M22 = 2.

Cofactors: A11 = (−1)^(1+1)M11 = +3,

A12 = (−1)^(1+2)M12 = −0 = 0,

A21 = (−1)^(2+1)M21 = −(−4) = 4,

A22 = (−1)^(2+2)M22 = +2.

(ii) For the determinant with elements a11=a, a12=c, a21=b, a22=d:

Minors: M11 = d, M12 = b, M21 = c, M22 = a.

Cofactors: A11 = d, A12 = −b, A21 = −c, A22 = a.

Answer: (i) M11=3, M12=0, M21=−4, M22=2; A11=3, A12=0, A21=4, A22=2. (ii) M11=d, M12=b, M21=c, M22=a; A11=d, A12=−b, A21=−c, A22=a.

Exercise 4.4

Find adjoint of each of the matrices: [[1, 2], [3, 4]]

A = [[1,2],[3,4]].

Cofactors: A11=4, A12=−3, A21=−2, A22=1.

adj A = [[A11, A21],[A12, A22]] = [[4, −2],[−3, 1]].

Equivalently: for a 2×2 matrix, interchange the diagonal entries and negate the off-diagonal entries.

Answer: adj A = [[4, −2], [−3, 1]]

All 61 questions in Determinants are solved step by step in the Solutions PDF above.

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